Prove that cot⁻¹(√(1+sin x) + √(1-sin x))/(√(1+sin x) - √(1-sin x)) = x/2; x ∈ (0, π/4)

LHS=cot⁻¹[(√(1+sin x) + √(1-sin x))/(√(1+sin x) - √(1-sin x))], x ∈ (0, π/4)
Given, 0 < x < π/4 ⇒ 0 < x < π/8 ⇒ x/2 ∈ (0, π/4) ⊂ (0, π)
cot⁻¹[ √((cos(x/2) + sin(x/2))²) + √((cos(x/2) - sin(x/2))²) / (√((cos(x/2) + sin(x/2))²) - √((cos(x/2) - sin(x/2))²))
= cot⁻¹[(cos(x/2) + sin(x/2) + cos(x/2) - sin(x/2)) / (cos(x/2) + sin(x/2) - cos(x/2) + sin(x/2))]
= cot⁻¹[2cos(x/2) / 2sin(x/2)]
⇒ cot⁻¹(cot(x/2))
⇒ x/2 = RHS
Hence proved.