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