If A+B+C= π than prove that cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

Steps to prove cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

L.H.S

=cosA+cosB+cosC

=2.cos(A+B)/2.cos(A-B)/2+(1-2sin^2C/2)

=2.sinC/2.cos(A-B)/2+1-2sin^2C/2 [sinC/2=cos(A+B)/2]

=2sinC/2{cos(A/2-B/2)-cos(A/2+B/2)}+1

=2sinC/2.2sinA/2.sinB/2+1

=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C

=R.H.S

Hence cosA+cosB+cosC=1+4.sin(1/2)A.sin(1/2)B.sin(1/2)C is proved.

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