Question

Please solve the question with all the steps !!

Answer 1

Answer:

LHS-

$\implies\:\sqrt\frac{cosec\theta\:-1}{cosec\theta\:+1} + \sqrt\frac{cosec\theta\:+1}{cosec\theta\:-1}$

$\implies\:\frac{\sqrt{cosec\theta\:-1}}{\sqrt{cosec\theta\:+1}} + \frac{\sqrt{cosec\theta\:+1}}{\sqrt{cosec\theta\:-1}}$

$\implies\:\frac{(\sqrt{cosec\theta\:-1} )^2 + \:(\sqrt{cosec\theta\:+1}) ^2}{(\sqrt{cosec\theta\:-1} ) \:(\sqrt{cosec\theta\:+1})}$

$\implies\:\frac{cosec\theta\:-1\:+cosec\theta\:+1}{\sqrt{cosec^2\theta\:-1} }$

We know that

1 + cot²Ф = cosec²Ф

∴ cot²Ф = cosec²Ф - 1

$\implies\:\frac{2cosec\theta}{\sqrt{cot^2\theta} }$

$\implies\:\frac{2cosec\theta}{cot\theta}$

$\implies\:\frac{2}{sin\theta}\frac{sin\theta}{cos\theta}$

$\implies\: 2sec\theta \implies\:R.H.S$