Prove the following identities: 1- ^2 ^2-1 = ^2 - Vidyadip

Question

Prove the following identities:
$\frac{1-\tan^2\theta}{\cot^2-1}=\tan^2\theta$

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Answer

$\text{LHS}=\frac{1-\tan^2\theta}{\cot^2\theta-1}$
$=\frac{1-\frac{\sin^2\theta}{\cos^2\theta}}{1+\frac{\sin^2\theta}{\cos^2\theta}}=\frac{{\begin{pmatrix}\frac{\cos^2\theta-\sin^2\theta}{\cos^2\theta}\end{pmatrix}}{}}{\begin{pmatrix}\frac{\cos^2\theta-\sin^2\theta}{\sin^2\theta}\end{pmatrix}}$
$=\Big(\frac{\cos^2\theta-\sin^2\theta}{\cos^2\theta}\Big)\times\frac{\sin^2\theta}{\big(\cos^2\theta-\sin^2\theta\big)}$
$=\frac{\sin^2\theta}{\cos^2\theta}=\tan^2\theta$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$

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