Question
Prove the following identites:
$\big(\sec^2\theta-1\big)\big(\text{cosec}^2\theta-1\big)=1$
$\big(\sec^2\theta-1\big)\big(\text{cosec}^2\theta-1\big)=1$
✓
Answer
$\text{L.H.S.}=\big(\sec^2\theta-1\big)\big(\text{cosec}^2\theta-1\big)$
$=\tan^2\theta\times\cot^2\theta$ $\begin{bmatrix}\because\big(\sec^2\theta-1\big)=\tan^2\theta,\\\big(\text{cosec}^2\theta-1\big)=\cot^2\theta\end{bmatrix}$
$=\tan^2\theta\times\frac{1}{\tan^2\theta}$
$=1$
$=\text{R.H.S.}$
$\therefore\text{LHS}=\text{RHS}$
$=\tan^2\theta\times\cot^2\theta$ $\begin{bmatrix}\because\big(\sec^2\theta-1\big)=\tan^2\theta,\\\big(\text{cosec}^2\theta-1\big)=\cot^2\theta\end{bmatrix}$
$=\tan^2\theta\times\frac{1}{\tan^2\theta}$
$=1$
$=\text{R.H.S.}$
$\therefore\text{LHS}=\text{RHS}$
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