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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 $$1+ \sec^2x + \frac{2}{\cos^2x \sin^2x}$$

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  • $\begingroup$ Please see math.meta.stackexchange.com/questions/5020 $\endgroup$ – Lord Shark the Unknown Dec 5 '18 at 20:34
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    $\begingroup$ 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
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(i) $$(\sec^2x + \tan^2x)(\csc^2x + \cot^2x)$$ => $$(\sec^2x\csc^2x + \tan^2x\csc^2x + \sec^2x\cot^2x + \tan^2x\cot^2x)$$ => $$(\sec^2x\csc^2x+ \frac{\sin^2x}{\cos^2x\sin^2x} + \frac{\cos^2x}{\cos^2x\sin^2x} + 1)$$ => $$(\sec^2x\csc^2x+ \frac{1}{\cos^2x} + \frac{1}{\sin^2x} + 1)$$ => $$(\sec^2x\csc^2x+ \frac{\sin^2x + \cos^2x}{\cos^2x\sin^2x} + 1)$$ => $$(\sec^2x\csc^2x+ \frac{1}{\cos^2x\sin^2x} + 1)$$ => $$(\sec^2x\csc^2x+ \sec^2x\csc^2x + 1)$$ => $$(1 + 2\sec^2x\csc^2x)$$

ii) $$\frac{\cos x}{1 - \tan x} + \frac{\sin x}{1 - \cot x}$$ => $$\frac{\cos x(1 - \cot x) + \sin x(1 - \tan x)}{(1 - \tan x)(1 - \cot x)}$$ => $$\frac{\frac{\cos^2x \sin x - \cos x\cos^2x + \sin^2 x\cos x - \sin x\sin^2x}{\sin x \cos x}}{(1 - \tan x)(1 - \cot x)}$$ => $$\frac{\cos^2 x(\sin x - \cos x) + \sin^2 x(\cos x - \sin x)}{\cos x\sin x(1 - \tan x)(1 - \cot x)}$$ => $$\frac{(\sin^2 x - \cos^2 x)(\cos x - \sin x)}{\cos x\sin x(1 - \tan x)(1 - \cot x)}$$ => $$\frac{(\sin^2 x - \cos^2x)(\cos x - \sin x)}{(\cos x - \sin x)(\sin x - \cos x)}$$ => $$\frac{(\sin^2 x - cos^2 x)}{(\sin x - \cos x)}$$ => $$\frac{(\sin x + \cos x)(\sin x - \cos x)}{(\sin x - \cos x)}$$ => $$\sin x + \cos x$$

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  • $\begingroup$ 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 $\frac{\cos x\sin x-\cos^2x}{\sin x}+\frac{\sin x\cos x-\sin^2x}{\cos x}$, rather than $\frac{\cos x\sin x-\cos^2x}{\sin x\cos x}+\frac{\sin x\cos x-\sin^2x}{\sin x\cos x}=\frac{\cos x\sin x-\cos^2x+\sin x\cos x-\sin^2x}{\sin x\cos x}$ 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
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$$(\sec^2x+\tan^2x)(\csc^2x+\cot^2x)$$

$$=(2\sec^2x-1)(2\csc^2x-1)$$

$$=4\sec^2x\csc^2x-2(\sec^2x+\csc^2x)+1$$

Now use $\sec^2x+\csc^2x=\cdots=\sec^2x\csc^2x$

The second one has been solved by Taussig

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  • $\begingroup$ Your method for this proof is definitely an improvement over mine. $\endgroup$ – N. F. Taussig Dec 8 '18 at 14:53
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(i)
\begin{align*} (\sec^2x + \tan^2x)(\csc^2x + \cot^2x) & = (1 + \tan^2x + \tan^2x)(1 + \cot^2x + \cot^2x)\\ & = (1 + 2\tan^2x)(1 + 2\cot^2x)\\ & = 1 + 2\cot^2x + 2\tan^2x + 4\\ & = 5 + 2(\csc^2x - 1) + 2(\sec^2x - 1)\\ & = 5 + 2\csc^2x - 2 + 2\sec^2x - 2\\ & = 1 + 2\csc^2x + 2\sec^2x\\ & = 1 + 2\left(\frac{1}{\sin^2x} + \frac{1}{\cos^2x}\right)\\ & = 1 + 2\left(\frac{\cos^2x + \sin^2x}{\sin^2x\cos^2x}\right)\\ & = 1 + \frac{2}{\cos^2x\sin^2x}\\ & = 1 + 2\sec^2x\csc^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*}

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(i) Let $C=\cos(2x), S=\sin(2x)=2\sin(x)\cos(x)$

$S^2*RHS= S^2(1+\frac{2}{(S/2)^2})= S^2+8=9-C^2$

$\begin{align}S^2*LHS &=(2+2\sin^2(x))(2+2\cos^2(x)) \cr &=(2+(1-C))(2+(1+C))\cr &=9-C^2 =S^2*RHS \end{align}$
QED

(ii) Let $t=\tan(x/2),\text{then }\sin(x)=\frac{2t}{1+t^2}\text{ , }\cos(x)=\frac{1-t^2}{1+t^2}$

$1-\tan(x)=1-\frac{2t}{1-t^2}=\frac{1-2t-t^2}{1-t^2}$
$1-\cot(x)=1-\frac{1-t^2}{2t}=\frac{1-2t-t^2}{-2t}$

$\begin{align} \frac{\cos(x)}{1-\tan(x)} + \frac{\sin(x)}{1-\cot(x)} &=\frac{(1-t^2)^2 - (2t)^2}{(1-2t-t^2)(1+t^2)} \cr &=\frac{(1-2t-t^2)(1+2t-t^2)}{(1-2t-t^2)(1+t^2)} \cr &= \frac{2t}{1+t^2} + \frac{1-t^2}{1+t^2} = \sin(x) + \cos(x) \end{align}$
QED

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