we have (sin A + cos A)^2 = sin ^2 A + 2 sin A cos A + cos ^2 A = 1 + 2 sin A cos A
(sin A - cos A)^2 = sin ^2 A - 2 sin A cos A + cos ^2 A = 1 - 2 sin A cos A
tan (pi/4-A) = (tan pi/4 - tan A)/(1 + tan pi/4 tan A) = (1- tan A )/(1+ tan A)
LHS =
(1 - sin x) / (1 + sin x)
= (1- 2 sin x/2 cos x/2)/(1+ 2 sin x/2 cos x/2)
= ( sin x/2- cos x/2)^2/(sin x/2+ cos x/2)^2 (from above)
= ((sin x/2- cos x/2)/ (sin x/2 + cos x/2))^2
= ((1- tan x/2)/(1+ tan x/2))^2
= ( tan (pi/4-x/2))^2 (from above)
proved
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