Prove that cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2)

Steps to prove cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2)

L.H.S

=cos^2(A/2)+cos^2(B/2)+cos^2(C/2)

=(1+cosA)/2+(1+cosB)/2-(1+cosC)/2

=(1+cosA+1+cosB-1-cosC)/2

=(1+cosA+cosB-cosC)/2

=1/2+(cosA+cosB-cosC)/2

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

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

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

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

=2.cos(A/2)cos(B/2).sin(C/2)

R.H.S

Hence cos^2(A/2)+cos^2(B/2)+cos^2(C/2)=2.cos(A/2)cos(B/2).sin(C/2) is proved.

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