Question

Prove the following:

`tan^-1[sqrt((1 - cosθ)/(1 + cosθ))] = θ/(2)`, if θ ∈ (– π, π).

Answer 1

`(1 - cosθ)/(1 + cosθ) = (2sin^2(θ/2))/(2cos^2(θ/2)`

= `tan^2(θ/2)`

∴ `sqrt((1 - cosθ)/(1 + cosθ)) = sqrt(tan^2(θ/2)`

= `tan(θ/2)`

∴ L.H.S. = `tan^-1[sqrt((1 - cosθ)/(1 + cosθ))]`

= `tan^-1[tan(θ/2)]`

= `θ/(2)`                      ...[∵ tan^–1(tan θ) = θ]

= R.H.S.