Question

Solve
$y=\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}$

Answer 1

$y=\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\times \left(\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{(\sqrt{1-\sin x}+\sqrt{1+\sin x})}\right)$

$=\frac{1-\sin x+1+\sin x+2\sqrt{1-{\sin }^{2}x}}{1-\sin x-1-\sin x}$

$=\frac{2+2\sqrt{{\cos }^{2}x}}{-2\sin x}$

$=\frac{1+\cos x}{-\sin x}$

$=\frac{2{\cos }^2\frac{x}{2}}{-2\sin \frac{x}{2}\cos \frac{x}{2}}$ since $\cos \theta =1-2{\cos }^2\frac{\theta }{2}$

$=\frac{-\cos \frac{x}{2}}{\sin \frac{x}{2}}$

$=-\cot \frac{x}{2}$