If A + B + C = pi then prove that sinA + sinB + sinC = 4cos(A/2) * cos(B/2) * cos(C/2)?

We have to prove sinA+sinB+sinC=4cos(A/2)*cos(B/2)*cos(C/2)

Solution

Let us start with LHS

sinA+sinB+sinC
=2sin(A+B)/2cos(A-B)/2+sin C
=2sin(pi-C)/2cos(A-B)/2+2sin C/2cosC/2
=2cosC/2(cos(A-B)/2+cos(A+B)/2)
=4 cos A/2 cos B/2 cos C/2

= RHS

Hence Proved

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