Question

I'm sure nobody can solve this trigonometric question​

Answer 1

L.H.S = √(1 - cos Ф)/√(1 + cos Ф)

rationalise by √(1 - cos Ф)

√(1 - cos Ф)² / √(1 - cos² Ф)

(1 - cos Ф) / √(sin² Ф)                          (sin² α + cos² α = 1 )

(1 - cos Ф) / sin Ф

1 / sin Ф - cos Ф / sin Ф

cosec Ф - cot Ф    

hence , L.H.S = R.H.S          

Answer 2

$\bf\large\underline\blue{Question:-}$

Prove that $\sqrt{ \frac{1 - \cos Q}{1 + \cos Q} } = \cosec Q - \cot Q$

$\bf\large\underline\blue{Solution:-}$

$\bf\underline\blue{Taking\:LHS:-}$

$\sqrt{ \frac{1 - \cos Q}{1 + \cos Q} }$

$= \sqrt{ \frac{1 - \cos Q}{1 + \cos Q} \times \frac{1 - \cos Q}{1 - \cos Q} }$

$= \sqrt{ \frac{ {(1 - \cos Q)}^{2} }{1 - \cos^{2}Q } }$

$= \sqrt{ \frac{ {(1 - \cos Q)}^{2} }{ \sin^{2} Q} }$

$= \frac{1 - \cos Q}{ \sin Q}$

$= \frac{1}{ \sin Q} - \frac{ \cos Q}{ \sin Q}$

$= \cosec Q - \cot Q$

$\bf\underline\blue{Taking\:RHS:-}$

$= \cosec Q - \cot Q$

$\bf\large\underline\blue{Therefore,}$

$\bf\underline\blue{LHS=RHS}$

$\bf\large\underline\blue{Hence\:Proved.}$