Question
Prove the following identites:
$\big(1+\cos\theta\big)\big(1-\cos\theta\big)\big(1+\cot^2\theta\big)=1$
$\big(1+\cos\theta\big)\big(1-\cos\theta\big)\big(1+\cot^2\theta\big)=1$
✓
Answer
$\big(1-\cos^2\theta\big)\big(1+\cot^2\theta\big)$
$=\sin^2\theta\times\text{cosec}^2\theta$ $\begin{bmatrix}\because\big(1-\cos^2\theta\big)=\sin^2\theta,\\\big(1+\cot^2\theta\big)=\text{cosec}^2\theta\end{bmatrix}$
$=\sin^2\theta\times\frac{1}{\sin^2\theta}$
$=1$
$=\text{R.H.S.}$
$=\sin^2\theta\times\text{cosec}^2\theta$ $\begin{bmatrix}\because\big(1-\cos^2\theta\big)=\sin^2\theta,\\\big(1+\cot^2\theta\big)=\text{cosec}^2\theta\end{bmatrix}$
$=\sin^2\theta\times\frac{1}{\sin^2\theta}$
$=1$
$=\text{R.H.S.}$
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