Ashley,

It is almost certainly true that the answer (while it exists) cannot be written in a closed form. [I assume that x is in radians here, by the way; but even if it is in degrees there almost certainly won't be a tidy solution.]

To get a rough approximation, plot y = sin(x) + cos(x) [if you get this right it should come out as a single sinusoidal function itself] and y=x, and see where they intersect. You should get exactly two solutions, not symmetric with respect to the origin.

To get a more accurate approximation use trial and error on a basic scientific calculator, a numerical solve function on a more advanced calculator, or Newton's method.

I'm curious where you got this problem from; it doesn't have a "nice" answer that would motivate putting it into a textbook, and mathematical models of real world problems would normally not have angles set equal to trig functions without a constant of some sort. I would expect instead to see something like sin x + cos x = ax.

Good Hunting!
RD