<s>[INST] 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?
 [/INST] $$\begin{align}
2\cdot \color{blue}{\cos^2\left(45^{\circ}-\frac{x}{2}\right)}&=\\
2\cdot \color{blue}{\frac{1+cos(double angle)}{2}}&=\\
2\cdot \color{blue}{\frac{1+cos(90^{\circ} - x)}{2}}&=\\
1+\cos\left(90^{\circ}-x\right) &= \\
1+\sin(x)&\\
\end{align}$$
</s>