Question 5
a) Prove that for every real number $x$ the arithmetic mean of $\sqrt{1 + \sin x}$ and $\sqrt{1 - \sin x}$ is equal to one of the following: $\sin \frac{x}{2}$, $\cos \frac{x}{2}$, $-\sin \frac{x}{2}$, $-\cos \frac{x}{2}$.
b) Can one leave out one of the four numbers listed in part a) in such a way that the claim still holds?
b) Can one leave out one of the four numbers listed in part a) in such a way that the claim still holds?