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Question
If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α
Solution
θ + Φ = α, tan θ = k tan Φ
k = `tantheta/tanphi`
`("k" - 1)/("k" + 1) = (tan theta/tan phi - 1)/(tan theta/tan phi + 1)`
= `(tan theta - tan phi)/(tan theta + tan phi)`
= `(sin theta/cos theta - sin phi/cos phi)/(sintheta/costheta + sin phi/cos phi)`
= `(sintheta cosphi - costheta sinphi)/(sintheta cosphi + costheta sin phi)`
`("k" - 1)/("k" + 1) = (sin(theta - phi))/(sin(theta + phi))`
= `(sin(theta - phi))/sin alpha`
`sin (theta - phi) = ("k" - 1)/("k" + 1) sin alpha`
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