If an angle α be divided into two parts such that the tangent of one part is λ times the tangent of the other part, prove that their difference θ is given by sin θ=λ−1λ+1sinα.
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Solution
Let α=A+B
Given that tanA=λtanB
Also, θ=A−B
Therefore, A=α+θ2;B=α−θ2
tanA=λtanB
tan(α+θ2)=λtan(α−θ2)
tan(α+θ2)tan(α−θ2)=λ
Converting into sin,cos and by applying componendo-dividendo, we get