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

tan⁻¹(cos x / (1 + sin x)) = tan⁻¹(cos²x/2 - sin²x/2)/(cos²x/2 + sin²x/2 + 2cos x/2 sin x/2) = tan⁻¹((cos x/2 - sin x/2)(cos x/2 + sin x/2)/(cos x/2 + sin x/2)² ) = tan⁻¹(cos x/2 - sin x/2 / cos x/2 + sin x/2) = tan⁻¹(1 - tan x/2 / 1 + tan x/2) = tan⁻¹(tan(π/4 - tan x/2) / (1 + tan(π/2) tan x/2)) = tan⁻¹[tan(π/4 - x/2)] = π/4 - x/2