Prove sin^2 (pi/4 + x/2)/sin^2(pi/4 - x/2) = (1+ sin x)/(1 - sin x)

The identity to be proved is [sin (pi/4 + x/2)]^2/[sin(pi/4 -
x/2)]^2 = (1+ sin x)/(1 - sin x)

We know that cos 2x = 1 – 2*(sin
x)^2

=> (sin x)^2 = (1 – cos
2x)/2

Let’s start with the left hand side of the given
identity

[sin(pi/4 + x/2)]^2/[sin(pi/4 -
x/2)]^2

=> [(1 – cos(pi/2 + x))/2]/[ (1 – cos(pi/2 -
x))/2]

use cos (pi/2 - x) = sin
x

=> ( 1 – (-sin x))/(1 - sin
x)

=> (1 + sin x)/(1 – sin
x)

which is the right hand
side.

This proves the identity sin^2 (pi/4 +
x/2)/sin^2(pi/4 - x/2) = (1+ sin x)/(1 - sin x)