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Question

If cosα+cosβ=12 and sinα+sinβ=13

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Solution

cosα+cosβ=1/2
2cos(α+β2)cos(αβ2)=12........(1)
sinα+sinβ=1/3
2sin(α+β2)cos(αβ2)=13........(2)
dividing (2) & (1), we get
tan(α+β2)=23
cos(α+β2)=±313
Substituting above in (1), we get
cos(αβ2)=±1312
tan(αβ2)=±13113

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