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Question

If mtan1(ab)a+btanθ2=cos1[acosθ+ba+bcosθ].Find m

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Solution

2tan1(aba+btanθ2)

2tan1x=cos1(1x21+x2)

=2tan1⎜ ⎜ ⎜aba+bsinθ2cosθ2⎟ ⎟ ⎟

=cos1⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜1(aba+b)⎜ ⎜ ⎜sin2θ2sin2θ2⎟ ⎟ ⎟1+(aba+b)⎜ ⎜ ⎜sin2θ2cos2θ2⎟ ⎟ ⎟⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

=cos1⎜ ⎜ ⎜(a+b)cos2θ2(ab)sin2θ2(a+b)cos2θ2+(ab)sin2θ2)⎟ ⎟ ⎟

=cos1⎜ ⎜ ⎜ ⎜a(cos2θ2sin2θ2)+b(cos2θ2+sin2θ2+sin2θ2)a(cos2θ2+sin2θ2)+b(cos2θ2sin2θ2)⎟ ⎟ ⎟ ⎟

=cos1(acosθ+ba+bcosθ)

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