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Question

In any triangles ABC prove the identities.
cosA+cosB+cosC=1+4sin(A/2)sin(B/2)sin(C/2).

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Solution

Here we want +1 and so we write
cosA=12sin2(A/2)
L.H.S. =12sin2(A/2)+2cos{(B+C)/2}cos{(BC)/2}
=12sin2(A/2)+2sin(A/2)cos{(BC)/2}
=12sin(A/2)[sin(A/2)cos{(BC)/2}]
=12sin(A/2)[cos{(B+C)/2}cos{(BC)/2}]
=12sin(A/2)[2sin(B/2)sin(C/2)]
=1+4sin(A/2)sin(B/2)sin(C/2)
sin(θ)=sinθ.

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