Prove that: (cos x) / (1 − sin x) = tan(π/4 + x/2)

LHS = cos x / (1 − sin x)

= sin(π/2 + x) / (1 + cos(π/2 + x))

[∵ sin A = 2 sin(A/2) cos(A/2)]

= (2 sin(π/4 + x/2) cos(π/4 + x/2)) / (2 cos²(π/4 + x/2))

[∵ cos A + 1 = 2 cos²(A/2)]

= tan(π/4 + x/2) = RHS