Proving $(sec^2x+tan^2x)(csc^2x+cot^2x)=1+2sec^2xcsc^2x$ and $frac{cos x}{1-tan x}+frac{sin x}{1-cot x} = sin...
$begingroup$
Prove the following identities:
$$(sec^2 x + tan^2x)(csc^2 x + cot^2x) = 1+ 2 sec^2x csc^2 x
tag i$$
$$frac{cos x}{1-tan x} + frac{sin x}{1-cot x} = sin x + cos x
tag {ii}$$
For $(mathrm i)$, I initially tried simplifying what was in the 2 brackets but ended up getting 1 + 1.
I then tried just multiplying out the brackets and got as far as $$1+ sec^2x + frac{2}{cos^2x sin^2x}$$
trigonometry
edited Dec 5 '18 at 23:51
Lorenzo B.
1,8402520
asked Dec 5 '18 at 20:27
S. BejtaS. Bejta
61
$endgroup$
add a comment |
$begingroup$
Prove the following identities:
$$(sec^2 x + tan^2x)(csc^2 x + cot^2x) = 1+ 2 sec^2x csc^2 x
tag i$$
$$frac{cos x}{1-tan x} + frac{sin x}{1-cot x} = sin x + cos x
tag {ii}$$
For $(mathrm i)$, I initially tried simplifying what was in the 2 brackets but ended up getting 1 + 1.
I then tried just multiplying out the brackets and got as far as $$1+ sec^2x + frac{2}{cos^2x sin^2x}$$
trigonometry
edited Dec 5 '18 at 23:51
Lorenzo B.
1,8402520
asked Dec 5 '18 at 20:27
S. BejtaS. Bejta
61
$endgroup$
$begingroup$
Please see math.meta.stackexchange.com/questions/5020
$endgroup$
– Lord Shark the Unknown
Dec 5 '18 at 20:34
1
$begingroup$
I've retyped your equations using mathJax. Notice you can further edit your question, as some parts of the equations were not clear, please do so if something is not as you originally intended.
$endgroup$
– Mefitico
Dec 5 '18 at 20:34
$begingroup$
Welcome to MathSE. Please edit the question to show your work. That helps us identify any errors you may have made. This tutorial explains how to typeset mathematics on this site. You should not have the $sec^2x$ term in your answer to the first question.
$endgroup$
– N. F. Taussig
Dec 5 '18 at 21:05
add a comment |
$begingroup$
Prove the following identities:
$$(sec^2 x + tan^2x)(csc^2 x + cot^2x) = 1+ 2 sec^2x csc^2 x
tag i$$
$$frac{cos x}{1-tan x} + frac{sin x}{1-cot x} = sin x + cos x
tag {ii}$$
For $(mathrm i)$, I initially tried simplifying what was in the 2 brackets but ended up getting 1 + 1.
I then tried just multiplying out the brackets and got as far as $$1+ sec^2x + frac{2}{cos^2x sin^2x}$$
trigonometry
edited Dec 5 '18 at 23:51
Lorenzo B.
1,8402520
asked Dec 5 '18 at 20:27
S. BejtaS. Bejta
61
$endgroup$
Prove the following identities:
$$(sec^2 x + tan^2x)(csc^2 x + cot^2x) = 1+ 2 sec^2x csc^2 x
tag i$$
$$frac{cos x}{1-tan x} + frac{sin x}{1-cot x} = sin x + cos x
tag {ii}$$
For $(mathrm i)$, I initially tried simplifying what was in the 2 brackets but ended up getting 1 + 1.
I then tried just multiplying out the brackets and got as far as $$1+ sec^2x + frac{2}{cos^2x sin^2x}$$
trigonometry
trigonometry
edited Dec 5 '18 at 23:51
Lorenzo B.
1,8402520
asked Dec 5 '18 at 20:27
S. BejtaS. Bejta
61
edited Dec 5 '18 at 23:51
Lorenzo B.
1,8402520
asked Dec 5 '18 at 20:27
S. BejtaS. Bejta
61
edited Dec 5 '18 at 23:51
Lorenzo B.
1,8402520
edited Dec 5 '18 at 23:51
Lorenzo B.
1,8402520
edited Dec 5 '18 at 23:51
Lorenzo B.
1,8402520
1,8402520
asked Dec 5 '18 at 20:27
S. BejtaS. Bejta
61
asked Dec 5 '18 at 20:27
S. BejtaS. Bejta
61
asked Dec 5 '18 at 20:27
S. BejtaS. Bejta
61
61
$begingroup$
Please see math.meta.stackexchange.com/questions/5020
$endgroup$
– Lord Shark the Unknown
Dec 5 '18 at 20:34
1
$begingroup$
I've retyped your equations using mathJax. Notice you can further edit your question, as some parts of the equations were not clear, please do so if something is not as you originally intended.
$endgroup$
– Mefitico
Dec 5 '18 at 20:34
$begingroup$
Welcome to MathSE. Please edit the question to show your work. That helps us identify any errors you may have made. This tutorial explains how to typeset mathematics on this site. You should not have the $sec^2x$ term in your answer to the first question.
$endgroup$
– N. F. Taussig
Dec 5 '18 at 21:05
add a comment |
$begingroup$
Please see math.meta.stackexchange.com/questions/5020
$endgroup$
– Lord Shark the Unknown
Dec 5 '18 at 20:34
1
$begingroup$
I've retyped your equations using mathJax. Notice you can further edit your question, as some parts of the equations were not clear, please do so if something is not as you originally intended.
$endgroup$
– Mefitico
Dec 5 '18 at 20:34
$begingroup$
Welcome to MathSE. Please edit the question to show your work. That helps us identify any errors you may have made. This tutorial explains how to typeset mathematics on this site. You should not have the $sec^2x$ term in your answer to the first question.
$endgroup$
– N. F. Taussig
Dec 5 '18 at 21:05
$begingroup$
Please see math.meta.stackexchange.com/questions/5020
$endgroup$
– Lord Shark the Unknown
Dec 5 '18 at 20:34
$begingroup$
Please see math.meta.stackexchange.com/questions/5020
$endgroup$
– Lord Shark the Unknown
Dec 5 '18 at 20:34
1
1
$begingroup$
I've retyped your equations using mathJax. Notice you can further edit your question, as some parts of the equations were not clear, please do so if something is not as you originally intended.
$endgroup$
– Mefitico
Dec 5 '18 at 20:34
$begingroup$
I've retyped your equations using mathJax. Notice you can further edit your question, as some parts of the equations were not clear, please do so if something is not as you originally intended.
$endgroup$
– Mefitico
Dec 5 '18 at 20:34
$begingroup$
Welcome to MathSE. Please edit the question to show your work. That helps us identify any errors you may have made. This tutorial explains how to typeset mathematics on this site. You should not have the $sec^2x$ term in your answer to the first question.
$endgroup$
– N. F. Taussig
Dec 5 '18 at 21:05
$begingroup$
Welcome to MathSE. Please edit the question to show your work. That helps us identify any errors you may have made. This tutorial explains how to typeset mathematics on this site. You should not have the $sec^2x$ term in your answer to the first question.
$endgroup$
– N. F. Taussig
Dec 5 '18 at 21:05
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
(i)
$$(sec^2x + tan^2x)(csc^2x + cot^2x)$$
=>
$$(sec^2xcsc^2x + tan^2xcsc^2x + sec^2xcot^2x + tan^2xcot^2x)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x}{cos^2xsin^2x} + frac{cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2x} + frac{1}{sin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x + cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ sec^2xcsc^2x + 1)$$
=>
$$(1 + 2sec^2xcsc^2x)$$
ii) $$frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x}$$
=>
$$frac{cos x(1 - cot x) + sin x(1 - tan x)}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{frac{cos^2x sin x - cos xcos^2x + sin^2 xcos x - sin xsin^2x}{sin x cos x}}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{cos^2 x(sin x - cos x) + sin^2 x(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2 x)(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2x)(cos x - sin x)}{(cos x - sin x)(sin x - cos x)}$$
=>
$$frac{(sin^2 x - cos^2 x)}{(sin x - cos x)}$$
=>
$$frac{(sin x + cos x)(sin x - cos x)}{(sin x - cos x)}$$
=>
$$sin x + cos x$$
answered Dec 5 '18 at 21:21
Gopal AnantharamanGopal Anantharaman
626
$endgroup$
$begingroup$
That's not the desired answer for (ii). Your only mistake that I can see is in going to the line with a fraction inside a fraction; in the numerator there, you don't have a common denominator. That is, the numerator should be $frac{cos xsin x-cos^2x}{sin x}+frac{sin xcos x-sin^2x}{cos x}$, rather than $frac{cos xsin x-cos^2x}{sin xcos x}+frac{sin xcos x-sin^2x}{sin xcos x}=frac{cos xsin x-cos^2x+sin xcos x-sin^2x}{sin xcos x}$ as you have it.
$endgroup$
– Toby Bartels
Dec 5 '18 at 21:34
$begingroup$
Thanks! fixed the issues...
$endgroup$
– Gopal Anantharaman
Dec 5 '18 at 21:56
$begingroup$
Well done I lost in that! I'll take a look to your work, Bye
$endgroup$
– gimusi
Dec 5 '18 at 22:02
$begingroup$
It looks good now.
$endgroup$
– Toby Bartels
Dec 6 '18 at 22:00
add a comment |
$begingroup$
$$(sec^2x+tan^2x)(csc^2x+cot^2x)$$
$$=(2sec^2x-1)(2csc^2x-1)$$
$$=4sec^2xcsc^2x-2(sec^2x+csc^2x)+1$$
Now use $sec^2x+csc^2x=cdots=sec^2xcsc^2x$
The second one has been solved by Taussig
answered Dec 6 '18 at 7:41
lab bhattacharjeelab bhattacharjee
224k15156274
$endgroup$
$begingroup$
Your method for this proof is definitely an improvement over mine.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 14:53
add a comment |
$begingroup$
(i)
begin{align*}
(sec^2x + tan^2x)(csc^2x + cot^2x) & = (1 + tan^2x + tan^2x)(1 + cot^2x + cot^2x)\
& = (1 + 2tan^2x)(1 + 2cot^2x)\
& = 1 + 2cot^2x + 2tan^2x + 4\
& = 5 + 2(csc^2x - 1) + 2(sec^2x - 1)\
& = 5 + 2csc^2x - 2 + 2sec^2x - 2\
& = 1 + 2csc^2x + 2sec^2x\
& = 1 + 2left(frac{1}{sin^2x} + frac{1}{cos^2x}right)\
& = 1 + 2left(frac{cos^2x + sin^2x}{sin^2xcos^2x}right)\
& = 1 + frac{2}{cos^2xsin^2x}\
& = 1 + 2sec^2xcsc^2x
end{align*}
(ii)
begin{align*}
frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x} & = frac{cos x}{1 - tan x} cdot frac{cos x}{cos x} + frac{sin x}{1 - cot x} cdot frac{sin x}{sin x}\
& = frac{cos^2x}{cos x - sin x} + frac{sin^2x}{sin x - cos x}\
& = frac{cos^2x}{cos x - sin x} - frac{sin^2x}{cos x - sin x}\
& = frac{cos^2x - sin^2x}{cos x - sin x}\
& = frac{(cos x + sin x)(cos x - sin x)}{cos x - sin x}\
& = cos x + sin x
end{align*}
answered Dec 5 '18 at 22:25
N. F. TaussigN. F. Taussig
43.9k93355
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3027602%2fproving-sec2x-tan2x-csc2x-cot2x-12-sec2x-csc2x-and-frac-cos-x%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
(i)
$$(sec^2x + tan^2x)(csc^2x + cot^2x)$$
=>
$$(sec^2xcsc^2x + tan^2xcsc^2x + sec^2xcot^2x + tan^2xcot^2x)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x}{cos^2xsin^2x} + frac{cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2x} + frac{1}{sin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x + cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ sec^2xcsc^2x + 1)$$
=>
$$(1 + 2sec^2xcsc^2x)$$
ii) $$frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x}$$
=>
$$frac{cos x(1 - cot x) + sin x(1 - tan x)}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{frac{cos^2x sin x - cos xcos^2x + sin^2 xcos x - sin xsin^2x}{sin x cos x}}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{cos^2 x(sin x - cos x) + sin^2 x(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2 x)(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2x)(cos x - sin x)}{(cos x - sin x)(sin x - cos x)}$$
=>
$$frac{(sin^2 x - cos^2 x)}{(sin x - cos x)}$$
=>
$$frac{(sin x + cos x)(sin x - cos x)}{(sin x - cos x)}$$
=>
$$sin x + cos x$$
answered Dec 5 '18 at 21:21
Gopal AnantharamanGopal Anantharaman
626
$endgroup$
$begingroup$
That's not the desired answer for (ii). Your only mistake that I can see is in going to the line with a fraction inside a fraction; in the numerator there, you don't have a common denominator. That is, the numerator should be $frac{cos xsin x-cos^2x}{sin x}+frac{sin xcos x-sin^2x}{cos x}$, rather than $frac{cos xsin x-cos^2x}{sin xcos x}+frac{sin xcos x-sin^2x}{sin xcos x}=frac{cos xsin x-cos^2x+sin xcos x-sin^2x}{sin xcos x}$ as you have it.
$endgroup$
– Toby Bartels
Dec 5 '18 at 21:34
$begingroup$
Thanks! fixed the issues...
$endgroup$
– Gopal Anantharaman
Dec 5 '18 at 21:56
$begingroup$
Well done I lost in that! I'll take a look to your work, Bye
$endgroup$
– gimusi
Dec 5 '18 at 22:02
$begingroup$
It looks good now.
$endgroup$
– Toby Bartels
Dec 6 '18 at 22:00
add a comment |
$begingroup$
(i)
$$(sec^2x + tan^2x)(csc^2x + cot^2x)$$
=>
$$(sec^2xcsc^2x + tan^2xcsc^2x + sec^2xcot^2x + tan^2xcot^2x)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x}{cos^2xsin^2x} + frac{cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2x} + frac{1}{sin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x + cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ sec^2xcsc^2x + 1)$$
=>
$$(1 + 2sec^2xcsc^2x)$$
ii) $$frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x}$$
=>
$$frac{cos x(1 - cot x) + sin x(1 - tan x)}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{frac{cos^2x sin x - cos xcos^2x + sin^2 xcos x - sin xsin^2x}{sin x cos x}}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{cos^2 x(sin x - cos x) + sin^2 x(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2 x)(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2x)(cos x - sin x)}{(cos x - sin x)(sin x - cos x)}$$
=>
$$frac{(sin^2 x - cos^2 x)}{(sin x - cos x)}$$
=>
$$frac{(sin x + cos x)(sin x - cos x)}{(sin x - cos x)}$$
=>
$$sin x + cos x$$
answered Dec 5 '18 at 21:21
Gopal AnantharamanGopal Anantharaman
626
$endgroup$
$begingroup$
That's not the desired answer for (ii). Your only mistake that I can see is in going to the line with a fraction inside a fraction; in the numerator there, you don't have a common denominator. That is, the numerator should be $frac{cos xsin x-cos^2x}{sin x}+frac{sin xcos x-sin^2x}{cos x}$, rather than $frac{cos xsin x-cos^2x}{sin xcos x}+frac{sin xcos x-sin^2x}{sin xcos x}=frac{cos xsin x-cos^2x+sin xcos x-sin^2x}{sin xcos x}$ as you have it.
$endgroup$
– Toby Bartels
Dec 5 '18 at 21:34
$begingroup$
Thanks! fixed the issues...
$endgroup$
– Gopal Anantharaman
Dec 5 '18 at 21:56
$begingroup$
Well done I lost in that! I'll take a look to your work, Bye
$endgroup$
– gimusi
Dec 5 '18 at 22:02
$begingroup$
It looks good now.
$endgroup$
– Toby Bartels
Dec 6 '18 at 22:00
add a comment |
$begingroup$
(i)
$$(sec^2x + tan^2x)(csc^2x + cot^2x)$$
=>
$$(sec^2xcsc^2x + tan^2xcsc^2x + sec^2xcot^2x + tan^2xcot^2x)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x}{cos^2xsin^2x} + frac{cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2x} + frac{1}{sin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x + cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ sec^2xcsc^2x + 1)$$
=>
$$(1 + 2sec^2xcsc^2x)$$
ii) $$frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x}$$
=>
$$frac{cos x(1 - cot x) + sin x(1 - tan x)}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{frac{cos^2x sin x - cos xcos^2x + sin^2 xcos x - sin xsin^2x}{sin x cos x}}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{cos^2 x(sin x - cos x) + sin^2 x(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2 x)(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2x)(cos x - sin x)}{(cos x - sin x)(sin x - cos x)}$$
=>
$$frac{(sin^2 x - cos^2 x)}{(sin x - cos x)}$$
=>
$$frac{(sin x + cos x)(sin x - cos x)}{(sin x - cos x)}$$
=>
$$sin x + cos x$$
answered Dec 5 '18 at 21:21
Gopal AnantharamanGopal Anantharaman
626
$endgroup$
(i)
$$(sec^2x + tan^2x)(csc^2x + cot^2x)$$
=>
$$(sec^2xcsc^2x + tan^2xcsc^2x + sec^2xcot^2x + tan^2xcot^2x)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x}{cos^2xsin^2x} + frac{cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2x} + frac{1}{sin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{sin^2x + cos^2x}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ frac{1}{cos^2xsin^2x} + 1)$$
=>
$$(sec^2xcsc^2x+ sec^2xcsc^2x + 1)$$
=>
$$(1 + 2sec^2xcsc^2x)$$
ii) $$frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x}$$
=>
$$frac{cos x(1 - cot x) + sin x(1 - tan x)}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{frac{cos^2x sin x - cos xcos^2x + sin^2 xcos x - sin xsin^2x}{sin x cos x}}{(1 - tan x)(1 - cot x)}$$
=>
$$frac{cos^2 x(sin x - cos x) + sin^2 x(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2 x)(cos x - sin x)}{cos xsin x(1 - tan x)(1 - cot x)}$$
=>
$$frac{(sin^2 x - cos^2x)(cos x - sin x)}{(cos x - sin x)(sin x - cos x)}$$
=>
$$frac{(sin^2 x - cos^2 x)}{(sin x - cos x)}$$
=>
$$frac{(sin x + cos x)(sin x - cos x)}{(sin x - cos x)}$$
=>
$$sin x + cos x$$
answered Dec 5 '18 at 21:21
Gopal AnantharamanGopal Anantharaman
626
edited Dec 5 '18 at 21:55
answered Dec 5 '18 at 21:21
Gopal AnantharamanGopal Anantharaman
626
answered Dec 5 '18 at 21:21
Gopal AnantharamanGopal Anantharaman
626
answered Dec 5 '18 at 21:21
Gopal AnantharamanGopal Anantharaman
626
626
$begingroup$
That's not the desired answer for (ii). Your only mistake that I can see is in going to the line with a fraction inside a fraction; in the numerator there, you don't have a common denominator. That is, the numerator should be $frac{cos xsin x-cos^2x}{sin x}+frac{sin xcos x-sin^2x}{cos x}$, rather than $frac{cos xsin x-cos^2x}{sin xcos x}+frac{sin xcos x-sin^2x}{sin xcos x}=frac{cos xsin x-cos^2x+sin xcos x-sin^2x}{sin xcos x}$ as you have it.
$endgroup$
– Toby Bartels
Dec 5 '18 at 21:34
$begingroup$
Thanks! fixed the issues...
$endgroup$
– Gopal Anantharaman
Dec 5 '18 at 21:56
$begingroup$
Well done I lost in that! I'll take a look to your work, Bye
$endgroup$
– gimusi
Dec 5 '18 at 22:02
$begingroup$
It looks good now.
$endgroup$
– Toby Bartels
Dec 6 '18 at 22:00
add a comment |
$begingroup$
That's not the desired answer for (ii). Your only mistake that I can see is in going to the line with a fraction inside a fraction; in the numerator there, you don't have a common denominator. That is, the numerator should be $frac{cos xsin x-cos^2x}{sin x}+frac{sin xcos x-sin^2x}{cos x}$, rather than $frac{cos xsin x-cos^2x}{sin xcos x}+frac{sin xcos x-sin^2x}{sin xcos x}=frac{cos xsin x-cos^2x+sin xcos x-sin^2x}{sin xcos x}$ as you have it.
$endgroup$
– Toby Bartels
Dec 5 '18 at 21:34
$begingroup$
Thanks! fixed the issues...
$endgroup$
– Gopal Anantharaman
Dec 5 '18 at 21:56
$begingroup$
Well done I lost in that! I'll take a look to your work, Bye
$endgroup$
– gimusi
Dec 5 '18 at 22:02
$begingroup$
It looks good now.
$endgroup$
– Toby Bartels
Dec 6 '18 at 22:00
$begingroup$
That's not the desired answer for (ii). Your only mistake that I can see is in going to the line with a fraction inside a fraction; in the numerator there, you don't have a common denominator. That is, the numerator should be $frac{cos xsin x-cos^2x}{sin x}+frac{sin xcos x-sin^2x}{cos x}$, rather than $frac{cos xsin x-cos^2x}{sin xcos x}+frac{sin xcos x-sin^2x}{sin xcos x}=frac{cos xsin x-cos^2x+sin xcos x-sin^2x}{sin xcos x}$ as you have it.
$endgroup$
– Toby Bartels
Dec 5 '18 at 21:34
$begingroup$
That's not the desired answer for (ii). Your only mistake that I can see is in going to the line with a fraction inside a fraction; in the numerator there, you don't have a common denominator. That is, the numerator should be $frac{cos xsin x-cos^2x}{sin x}+frac{sin xcos x-sin^2x}{cos x}$, rather than $frac{cos xsin x-cos^2x}{sin xcos x}+frac{sin xcos x-sin^2x}{sin xcos x}=frac{cos xsin x-cos^2x+sin xcos x-sin^2x}{sin xcos x}$ as you have it.
$endgroup$
– Toby Bartels
Dec 5 '18 at 21:34
$begingroup$
Thanks! fixed the issues...
$endgroup$
– Gopal Anantharaman
Dec 5 '18 at 21:56
$begingroup$
Thanks! fixed the issues...
$endgroup$
– Gopal Anantharaman
Dec 5 '18 at 21:56
$begingroup$
Well done I lost in that! I'll take a look to your work, Bye
$endgroup$
– gimusi
Dec 5 '18 at 22:02
$begingroup$
Well done I lost in that! I'll take a look to your work, Bye
$endgroup$
– gimusi
Dec 5 '18 at 22:02
$begingroup$
It looks good now.
$endgroup$
– Toby Bartels
Dec 6 '18 at 22:00
$begingroup$
It looks good now.
$endgroup$
– Toby Bartels
Dec 6 '18 at 22:00
add a comment |
$begingroup$
$$(sec^2x+tan^2x)(csc^2x+cot^2x)$$
$$=(2sec^2x-1)(2csc^2x-1)$$
$$=4sec^2xcsc^2x-2(sec^2x+csc^2x)+1$$
Now use $sec^2x+csc^2x=cdots=sec^2xcsc^2x$
The second one has been solved by Taussig
answered Dec 6 '18 at 7:41
lab bhattacharjeelab bhattacharjee
224k15156274
$endgroup$
$begingroup$
Your method for this proof is definitely an improvement over mine.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 14:53
add a comment |
$begingroup$
$$(sec^2x+tan^2x)(csc^2x+cot^2x)$$
$$=(2sec^2x-1)(2csc^2x-1)$$
$$=4sec^2xcsc^2x-2(sec^2x+csc^2x)+1$$
Now use $sec^2x+csc^2x=cdots=sec^2xcsc^2x$
The second one has been solved by Taussig
answered Dec 6 '18 at 7:41
lab bhattacharjeelab bhattacharjee
224k15156274
$endgroup$
$begingroup$
Your method for this proof is definitely an improvement over mine.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 14:53
add a comment |
$begingroup$
$$(sec^2x+tan^2x)(csc^2x+cot^2x)$$
$$=(2sec^2x-1)(2csc^2x-1)$$
$$=4sec^2xcsc^2x-2(sec^2x+csc^2x)+1$$
Now use $sec^2x+csc^2x=cdots=sec^2xcsc^2x$
The second one has been solved by Taussig
answered Dec 6 '18 at 7:41
lab bhattacharjeelab bhattacharjee
224k15156274
$endgroup$
$$(sec^2x+tan^2x)(csc^2x+cot^2x)$$
$$=(2sec^2x-1)(2csc^2x-1)$$
$$=4sec^2xcsc^2x-2(sec^2x+csc^2x)+1$$
Now use $sec^2x+csc^2x=cdots=sec^2xcsc^2x$
The second one has been solved by Taussig
answered Dec 6 '18 at 7:41
lab bhattacharjeelab bhattacharjee
224k15156274
answered Dec 6 '18 at 7:41
lab bhattacharjeelab bhattacharjee
224k15156274
answered Dec 6 '18 at 7:41
lab bhattacharjeelab bhattacharjee
224k15156274
answered Dec 6 '18 at 7:41
lab bhattacharjeelab bhattacharjee
224k15156274
224k15156274
$begingroup$
Your method for this proof is definitely an improvement over mine.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 14:53
add a comment |
$begingroup$
Your method for this proof is definitely an improvement over mine.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 14:53
$begingroup$
Your method for this proof is definitely an improvement over mine.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 14:53
$begingroup$
Your method for this proof is definitely an improvement over mine.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 14:53
add a comment |
$begingroup$
(i)
begin{align*}
(sec^2x + tan^2x)(csc^2x + cot^2x) & = (1 + tan^2x + tan^2x)(1 + cot^2x + cot^2x)\
& = (1 + 2tan^2x)(1 + 2cot^2x)\
& = 1 + 2cot^2x + 2tan^2x + 4\
& = 5 + 2(csc^2x - 1) + 2(sec^2x - 1)\
& = 5 + 2csc^2x - 2 + 2sec^2x - 2\
& = 1 + 2csc^2x + 2sec^2x\
& = 1 + 2left(frac{1}{sin^2x} + frac{1}{cos^2x}right)\
& = 1 + 2left(frac{cos^2x + sin^2x}{sin^2xcos^2x}right)\
& = 1 + frac{2}{cos^2xsin^2x}\
& = 1 + 2sec^2xcsc^2x
end{align*}
(ii)
begin{align*}
frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x} & = frac{cos x}{1 - tan x} cdot frac{cos x}{cos x} + frac{sin x}{1 - cot x} cdot frac{sin x}{sin x}\
& = frac{cos^2x}{cos x - sin x} + frac{sin^2x}{sin x - cos x}\
& = frac{cos^2x}{cos x - sin x} - frac{sin^2x}{cos x - sin x}\
& = frac{cos^2x - sin^2x}{cos x - sin x}\
& = frac{(cos x + sin x)(cos x - sin x)}{cos x - sin x}\
& = cos x + sin x
end{align*}
answered Dec 5 '18 at 22:25
N. F. TaussigN. F. Taussig
43.9k93355
$endgroup$
add a comment |
$begingroup$
(i)
begin{align*}
(sec^2x + tan^2x)(csc^2x + cot^2x) & = (1 + tan^2x + tan^2x)(1 + cot^2x + cot^2x)\
& = (1 + 2tan^2x)(1 + 2cot^2x)\
& = 1 + 2cot^2x + 2tan^2x + 4\
& = 5 + 2(csc^2x - 1) + 2(sec^2x - 1)\
& = 5 + 2csc^2x - 2 + 2sec^2x - 2\
& = 1 + 2csc^2x + 2sec^2x\
& = 1 + 2left(frac{1}{sin^2x} + frac{1}{cos^2x}right)\
& = 1 + 2left(frac{cos^2x + sin^2x}{sin^2xcos^2x}right)\
& = 1 + frac{2}{cos^2xsin^2x}\
& = 1 + 2sec^2xcsc^2x
end{align*}
(ii)
begin{align*}
frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x} & = frac{cos x}{1 - tan x} cdot frac{cos x}{cos x} + frac{sin x}{1 - cot x} cdot frac{sin x}{sin x}\
& = frac{cos^2x}{cos x - sin x} + frac{sin^2x}{sin x - cos x}\
& = frac{cos^2x}{cos x - sin x} - frac{sin^2x}{cos x - sin x}\
& = frac{cos^2x - sin^2x}{cos x - sin x}\
& = frac{(cos x + sin x)(cos x - sin x)}{cos x - sin x}\
& = cos x + sin x
end{align*}
answered Dec 5 '18 at 22:25
N. F. TaussigN. F. Taussig
43.9k93355
$endgroup$
add a comment |
$begingroup$
(i)
begin{align*}
(sec^2x + tan^2x)(csc^2x + cot^2x) & = (1 + tan^2x + tan^2x)(1 + cot^2x + cot^2x)\
& = (1 + 2tan^2x)(1 + 2cot^2x)\
& = 1 + 2cot^2x + 2tan^2x + 4\
& = 5 + 2(csc^2x - 1) + 2(sec^2x - 1)\
& = 5 + 2csc^2x - 2 + 2sec^2x - 2\
& = 1 + 2csc^2x + 2sec^2x\
& = 1 + 2left(frac{1}{sin^2x} + frac{1}{cos^2x}right)\
& = 1 + 2left(frac{cos^2x + sin^2x}{sin^2xcos^2x}right)\
& = 1 + frac{2}{cos^2xsin^2x}\
& = 1 + 2sec^2xcsc^2x
end{align*}
(ii)
begin{align*}
frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x} & = frac{cos x}{1 - tan x} cdot frac{cos x}{cos x} + frac{sin x}{1 - cot x} cdot frac{sin x}{sin x}\
& = frac{cos^2x}{cos x - sin x} + frac{sin^2x}{sin x - cos x}\
& = frac{cos^2x}{cos x - sin x} - frac{sin^2x}{cos x - sin x}\
& = frac{cos^2x - sin^2x}{cos x - sin x}\
& = frac{(cos x + sin x)(cos x - sin x)}{cos x - sin x}\
& = cos x + sin x
end{align*}
answered Dec 5 '18 at 22:25
N. F. TaussigN. F. Taussig
43.9k93355
$endgroup$
(i)
begin{align*}
(sec^2x + tan^2x)(csc^2x + cot^2x) & = (1 + tan^2x + tan^2x)(1 + cot^2x + cot^2x)\
& = (1 + 2tan^2x)(1 + 2cot^2x)\
& = 1 + 2cot^2x + 2tan^2x + 4\
& = 5 + 2(csc^2x - 1) + 2(sec^2x - 1)\
& = 5 + 2csc^2x - 2 + 2sec^2x - 2\
& = 1 + 2csc^2x + 2sec^2x\
& = 1 + 2left(frac{1}{sin^2x} + frac{1}{cos^2x}right)\
& = 1 + 2left(frac{cos^2x + sin^2x}{sin^2xcos^2x}right)\
& = 1 + frac{2}{cos^2xsin^2x}\
& = 1 + 2sec^2xcsc^2x
end{align*}
(ii)
begin{align*}
frac{cos x}{1 - tan x} + frac{sin x}{1 - cot x} & = frac{cos x}{1 - tan x} cdot frac{cos x}{cos x} + frac{sin x}{1 - cot x} cdot frac{sin x}{sin x}\
& = frac{cos^2x}{cos x - sin x} + frac{sin^2x}{sin x - cos x}\
& = frac{cos^2x}{cos x - sin x} - frac{sin^2x}{cos x - sin x}\
& = frac{cos^2x - sin^2x}{cos x - sin x}\
& = frac{(cos x + sin x)(cos x - sin x)}{cos x - sin x}\
& = cos x + sin x
end{align*}
answered Dec 5 '18 at 22:25
N. F. TaussigN. F. Taussig
43.9k93355
answered Dec 5 '18 at 22:25
N. F. TaussigN. F. Taussig
43.9k93355
answered Dec 5 '18 at 22:25
N. F. TaussigN. F. Taussig
43.9k93355
answered Dec 5 '18 at 22:25
N. F. TaussigN. F. Taussig
43.9k93355
43.9k93355
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3027602%2fproving-sec2x-tan2x-csc2x-cot2x-12-sec2x-csc2x-and-frac-cos-x%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Please see math.meta.stackexchange.com/questions/5020
$endgroup$
– Lord Shark the Unknown
Dec 5 '18 at 20:34
1
$begingroup$
I've retyped your equations using mathJax. Notice you can further edit your question, as some parts of the equations were not clear, please do so if something is not as you originally intended.
$endgroup$
– Mefitico
Dec 5 '18 at 20:34
$begingroup$
Welcome to MathSE. Please edit the question to show your work. That helps us identify any errors you may have made. This tutorial explains how to typeset mathematics on this site. You should not have the $sec^2x$ term in your answer to the first question.
$endgroup$
– N. F. Taussig
Dec 5 '18 at 21:05