from $1-\sin x $ to $2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$ How can you go from $1-\sin x $  to  $2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$? I mean how to prove that $1-\sin x  = 2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$?

Use $\displaystyle \sin x=\cos\left(\frac\pi2-x\right)$ and  $\displaystyle\cos2y=1-2\sin^2y$
