Question

Please answer step by step with attachment

Answer 1

To prove:

$\sf{\frac{1}{1-cos\theta}-\frac{1}{1+cos\theta}=2cot\theta.cosec\theta}$

Proof:

$\sf{L.H.S.=\frac{1}{1-cos\theta}-\frac{1}{1+cos\theta}}$

$\sf{=\frac{(1+cos\theta)-(1-cos\theta)}{(1-cos\theta)(1+cos\theta)}}$

$\sf{=\frac{2cos\theta}{1-cos^{2}\theta}}$

$\sf\blue{Trigonometric \ identity}$

$\sf\blue{1-cos^{2}\theta=sin^{2}\theta}$

$\sf{=\frac{2cos\theta}{sin^{2}\theta}}$

$\sf{=\frac{2cos\theta}{sin\theta}\times\frac{1}{sin\theta}}$

$\sf\blue{Trigonometric \ ratio}$

$\sf\blue{\frac{cos\theta}{sin\theta}=cot\theta \ and \ \frac{1}{sin\theta}=cosec\theta}$

$\sf{=2cot\theta.cosec\theta}$

$\sf{=R.H.S.}$

$\sf\purple{\therefore{\frac{1}{1-cos\theta}-\frac{1}{1+cos\theta}=2cot\theta.cosec\theta}}$

$\sf{Hence, \ proved.}$