Prove the following trigonometric identities. 1- 1+ =cosec - - Vidyadip

Question

Prove the following trigonometric identities.
$\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}=\text{cosec}\theta-\cot\theta$

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Answer

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

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