Question

Prove that:
2-2cos(x-y)=4sin²{(x-y)/2}

Answer 1

We know that cos2x=cos(x+x)=cos^{2}x-sin^{2}x and sin^{2}x+cos^{2}x=1
so, cos2x = 1-2sin^{2}x or 1-cos2x=2sin^{2}x

LHS:
2-2cos(x-y)=2{1-cos(x-y)}= 2{1-cos2((x-y)/2}
                  =2{2sin^{2}((x-y)/2}
                  =4sin^2{(x-y)/2} = RHS
Hence proved.