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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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)

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