38. (4) Find all functions $f:[-2 ; 2] \rightarrow \mathbb{R}$ such that for any real $x$ and $y$ the equality $f(\sin x +$ $\cos y) + f(\sin y + \cos x) = \sin (x + y)$ holds.
Solution. Substitute $y=\pi / 2+x$, we get $f(0)+f(2 \cos x)=\cos 2 x=2 \cos ^{2} x-1$, from which $f(t)=t^{2} / 2-(f(0)+1)$ for any $t \in[-2,2]$. Substituting $t=0$ we find $f(0)=-1 / 2, f(t)=t^{2} / 2-1 / 2$. It is easy to see that such a function fits.