Answer
$(\cos \frac{x}{2}-\sin\frac{x}{2})^2=1-\sin x$
Work Step by Step
Start with the left side:
$(\cos \frac{x}{2}-\sin\frac{x}{2})^2$
Expand:
$=\cos ^2\frac{x}{2}-2\cos \frac{x}{2}\sin\frac{x}{2}+\sin^2\frac{x}{2}$
Rearrange terms:
$=\cos ^2\frac{x}{2}+\sin^2\frac{x}{2}-2\cos \frac{x}{2}\sin\frac{x}{2}$
Use the identities $\cos u+\sin u=1$ (where $u=\frac{x}{2}$) and $\sin 2u=2\sin u\cos u$ (where $u=\frac{x}{2}$):
$=1-\sin (2*\frac{x}{2})$
$=1-\sin x$
Since this equals the right side, the identity is proven.
