Answer
$(\sin(\frac{x}{2})+\cos(\frac{x}{2}))^2=1+\sin x$
Work Step by Step
Start with the left side:
$(\sin(\frac{x}{2})+\cos(\frac{x}{2}))^2$
Expand:
$=\sin^2(\frac{x}{2})+2\sin(\frac{x}{2})\cos(\frac{x}{2})+\cos^2(\frac{x}{2})$
$=\sin^2(\frac{x}{2})+\cos^2(\frac{x}{2})+2\sin(\frac{x}{2})\cos(\frac{x}{2})$
$=1+2\sin(\frac{x}{2})\cos(\frac{x}{2})$
Use the Double-Angle Formula $\sin(2u)=2\sin u\cos u$ where $u=\frac{x}{2}$:
$=1+\sin(2*\frac{x}{2})$
$=1+\sin x$
Since this equals the right side, the identity has been proven.
