parmenides51 wrote: Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$. Let $P(x)$ be the assertion $f(\sin x)+af(\cos x)=\cos 2x$ If $a\ne 1$, $f(x)=\frac{1-2x^2}{1-a}$ fits (it is not the only solution) If $a=1$ : subtracting $P(x)$ from $P(\frac{\pi}2-x)$ gives $\cos 2x=0\quad\forall x$, impossible Hence the answer $\boxed{a\in\mathbb R\setminus\{1\}}$