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Question
Prove the following:
`tan^-1[sqrt((1 - cosθ)/(1 + cosθ))] = θ/(2)`, if θ ∈ (– π, π).
Solution
`(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.
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