Question
Prove the following identities:
$\big(1+\tan^2\theta\big)\big(1+\cot^2\theta\big)=\frac{1}{\big(\sin^2\theta-\sin^4\theta\big)}$
$\big(1+\tan^2\theta\big)\big(1+\cot^2\theta\big)=\frac{1}{\big(\sin^2\theta-\sin^4\theta\big)}$
✓
Answer
$\text{L.H.S.}=\big(1+\tan^2\theta\big)\big(1+\cot^2\theta\big)$
$=\sec^2\theta\text{ cosec}^2\theta$
$=\frac{1}{\sin^2\theta\cos^2\theta}=\frac{1}{\sin^2\theta\big(1-\sin^2\theta\big)}$
$=\frac{1}{\sin^2\theta-\sin^4\theta}$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$
$=\sec^2\theta\text{ cosec}^2\theta$
$=\frac{1}{\sin^2\theta\cos^2\theta}=\frac{1}{\sin^2\theta\big(1-\sin^2\theta\big)}$
$=\frac{1}{\sin^2\theta-\sin^4\theta}$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$
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