4
$\begingroup$

a) Suppose that $E(x)$ is an even function and that $O(x)$ is an odd function. Suppose furthermore that $E(x) + O(x) = 0$. Show that for all $x$, $E(x) = 0$ and $O(x) = 0$.

b) Use part a) to show that if $A\sin(ax)+B\cos(bx) = 0$ for all $x$, where $A,B,a,b$ are fixed real numbers, then $B = 0$ and one of either $A$ or $a$ is also equal to $0$.

3 Answers 3