Prove the following identities: 1- 1+ =(cosec + ) - Vidyadip

Question

Prove the following identities:
$\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}=(\text{cosec}\theta+\cot\theta)$

✓

Answer

$\text{LHS}=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$
$=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}\times\frac{1-\cos\theta}{1-\cos\theta}}$
$=\sqrt{\frac{(1-\cos\theta)^2}{1-\cos^2\theta}}$
$=\sqrt{\frac{(1-\cos\theta)^2}{\sin^2\theta}}$
$=\frac{1+\cos\theta}{\sin\theta}$
$=\frac{1}{\sin\theta}+\frac{\cos\theta}{\sin\theta}$
$=\text{cosec}\theta-\cot\theta$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$

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