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Question

Show that:
cos1[cosα+cosβ1+cosαcosβ]=2tan1(tanα2tanβ2)

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Solution

cos1[cosα+cosβ1+cosαcosβ]=2tan1(tanα2tanβ2)
RHS =2tan1[tanα2tanβ2]
=cos1[1tan2α/2tan2β/21+tan2α/2tan2β/2]
=cos1⎢ ⎢ ⎢ ⎢ ⎢1sin2α/2sin2β/2cos2α/2cos2β/21+sin2α/2sin2β/2cos2α/2cos2β/2⎥ ⎥ ⎥ ⎥ ⎥
=cos1[cos2α/2cos2β/2sin2α/2sin2β/2cos2α/2cos2β/2+sin2α/2sin2β/2]
=cos1[2cos2α/2cos2β/2sin2α/2sin2β/22cos2α/2cos2β/2+2sin2α/2sin2β/2]
=cos1[(1+cosα)(1+cosβ)(1cosα)(1cosβ)(1+cosα)(1+cosβ)+(1cosα)(1cosβ)]
=cos1[cosα+cosβ1cosαcosβ]
=LHS

1129344_872312_ans_f04af259706d4b2dbf18f62a9e8e66c5.jpg

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