trigonometry - How should I go about finding angles at which two trigonometric ratios are equal?

Let's say we have an equation, $\sin x = \cos x$. We know that between $0^o \text{and} \ 90^o$, $\sin$ and $\cos$ are equal for $x = 45^o$. This is something we can easily figure because we memorize the trig ratios for angles between 0 and 90. How should I go about finding other angles for which an equality between two ratios hold?

I found by trial and error that this equation is true for $x = 225^o$-- but I'd like to if there is a method to do it without trial and error and for all other trigonometric ratios as well (for example, say, $\tan x = \cot x$).