Question

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

Answer 1

LHS

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

Dividing and multiplying by $\cos x$

$\frac{\sec x+1+\tan x}{\sec x+1-\tan x}$

$\frac{\sec x+\tan x+{\sec }^{2}x-{\tan }^{2}x}{\sec x+1-\tan x}$

$\frac{\sec x+\tan x+(\sec x-\tan x)(\sec x+\tan x)}{\sec x+1-\tan x}$

$\frac{\sec x+\tan x(\sec x-\tan x+1)}{\sec x+1-\tan x}$

$\sec x+\tan x$

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