from $1-\sin x $ to $2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$

Question

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 $1-\sin x = 2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$?

Answer 1

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