Question

Prove the following:

`tan^-1["cosθ + sinθ"/"cosθ - sinθ"] = pi/(4) + θ, if θ ∈ (- pi/4, pi/4)`

Answer 1

L.H.S. = `tan^-1["cosθ + sinθ"/"cosθ - sinθ"]`

= `tan^-1 [((cosθ)/(cosθ) + (sinθ)/(cosθ))/((cosθ)/(cosθ) - (sinθ)/(cosθ))]`

= `tan^-1((1 + tanθ)/(1 - tanθ))`

= `tan^-1[(tan  pi/4 + tanθ)/(1 - tan  pi/4 tan θ)]`

= `tan^-1[tan(pi/4 + θ)]`

= `pi/(4) + θ`           ...[∵ tan^–1(tan θ) = θ]
= R.H.S.