Prove the identity $1 + \sin x = 2 \cos^2 \left(45° - \frac{x}{2}\right)$ Here is the problem:
                     $$1 + \sin x = 2 \cos^2 \left(45° - \frac{x}{2}\right)$$
Can you help me prove  that this is an trigonometric identity?

$$1 + \sin x = 2 \cos^2 \left(45° - \frac{x}{2}\right)$$
$$\cos{2\alpha}=2cos^{2}{\alpha}-1$$
$$\cos{2\cdot \left(45^{\circ}-\frac{x}{2}\right)}=\cos{\left(90^{\circ}-x\right)}=\cos{90^{\circ}\cos{x}}+\sin{90^{\circ}\sin{x}}=0+1\cdot \sin{x}=\sin{x}.$$
