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