Prove this identity using the product-to-sum identity for sine: sin^2x=(1-cos(2x))/(2) umm pls

Question

Prove this identity using the product-to-sum identity for sine:


`sin^2x=(1-cos(2x))/(2)`
umm pls

Answer 1

It follows from

`\cos2x=\cos(x+x)=\cos x\cos x-\sin x\sin x=\cos^2x-\sin^2x=1-2\sin^2x`

`\implies\cos 2x-1=-2\sin^2x`

`\implies\sin^2x=\frac{1-\cos2x}2`