Question
Prove the following identites:
$\big(1+\cot^2\theta\big)\sin^2\theta=1$
$\big(1+\cot^2\theta\big)\sin^2\theta=1$
✓
Answer
$\text{L.H.S.}=\big(1+\cot^2\theta\big)\sin^2\theta$
$=\text{cosec}^2\times\sin^2\theta$ $\Big[\because\big(1+\cot^2\theta\big)=\text{cosec}^2\theta\Big]$
$=\frac{1}{\sin^2\theta}\times\sin^2\theta$
$=1$
$=\text{R.H.S.}$
$\therefore\text{LHS}=\text{RHS}$
$=\text{cosec}^2\times\sin^2\theta$ $\Big[\because\big(1+\cot^2\theta\big)=\text{cosec}^2\theta\Big]$
$=\frac{1}{\sin^2\theta}\times\sin^2\theta$
$=1$
$=\text{R.H.S.}$
$\therefore\text{LHS}=\text{RHS}$
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