Question

Solve:
$\sqrt{\frac{\sin \left(x-\alpha \right)}{\sin \left(x+\alpha \right)}}$

Answer 1

We have,

$\sqrt{\frac{\sin (x-0)}{\sin (x+0)}}$

on rationalize and we get,

$=\sqrt{\frac{\sin (x-a)}{\sin (x+a)}}\times \sqrt{\frac{\sin (x-a)}{\sin (x-a)}}$

$=\sqrt{\frac{{\sin }^2(x-a)}{\sin (x+a)\sin (x-a)}}$

$=\frac{\sin (x-a)}{\sin (x+a)}$

$=\frac{\sin (x-a)}{\sqrt{{\sin }^{2}x-{\sin }^{2}a}}[\therefore \sin (A+B)\sin (A-B)={\sin }^{2}A-{\sin }^{2}B]$

Hence this is the answer.