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Question

Prove that
tan1(13tanx2)=12cos1(1+2cosx2+cosx)

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Solution

tan1(13)tan(x2)=(12)2tan113tan1(13)tan(x2)

=(12)cos1[1(13)tan2x21+(13)tan2x2]

=(12)cos1[3tan²(x/2)/3+tan²(x/2)][3tan²(x2)tan²(x2)]

=(12)cos13(1cosx)((1+cosx)3)+[(1cosx)(1+cosx)]

=(12)cos1[(2+4cosx)(4+2cosx)]

=(12)cos1[(1+2cosx)(2+cosx)]

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