Question
Prove the following identities:
$\frac{\tan^2\theta}{\big(1+\tan^2\theta\big)}+\frac{\cot^2\theta}{\big(1+\cot^2\theta\big)}=1$

Answer

$\text{L.H.S.}=\frac{\tan^2\theta}{\big(1+\tan^2\theta\big)}+\frac{\cot^2\theta}{\big(1+\cot^2\theta\big)}$
$=\frac{\tan^2\theta}{\sec^2\theta}+\frac{\cot^2\theta}{\text{cosec}^2\theta}$ $\Big[\because\big(1+\tan^2\theta\big)=\sec^2\theta$ and $\big(1+\cot^2\theta\big)=\text{cosec}^2\theta\Big]$
$=\frac{\frac{\sin^2\theta}{\cos^2\theta}}{\frac{1}{\cos^2\theta}}+\frac{\frac{\cos^2\theta}{\sin^2\theta}}{\frac{1}{\sin^2\theta}}$
$=\sin^2\theta+\cos^2\theta=1$
$=\text{R.H.S.}$
Hence, LHS = RHS.

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