Question

Prove that: $\sqrt{\frac{cosecx-1}{cosecx+1}}=\frac{1}{secx+tanx}$

Answer 1

$\sqrt{\frac{cosecx-1}{\csc +1}}$

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

$=\sqrt{\frac{(1-\sin x)(1-\sin x)}{(1+\sin x)(1-\sin x)}}$

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

$=\frac{1-\sin x}{\cos x}$ ...........$[sin²\theta +cos²\theta =1]$

$=\sec x-\tan x$

$=\frac{(\sec x-\tan x)(\sec x+\tan x)}{\sec x+\tan x}$

$=\frac{{\sec }^{2}x-{\tan }^{2}x)}{\sec x+\tan x}$

$=\frac{1}{\sec x+\tan x}$ ........... ${\sec }^2\theta =1+{\tan }^2\theta$