How to find intersections between two trig equations

Like cos2x = sinx.
is there a way to find it out WITHOUT graphing it?

Use the identity
cos 2x = 1 - 2 (sinx)^2

So your problem will be a quadratic in sinx:

1- 2(sinx)^2 = sinx

=> 2(sinx)^2 +sinx-1 = 0

From the quadratic formula:

sinx = -1/4 + sqrt(1+8)/4 or sinx = -1/4 - sqrt(1+8)/4
i.e. sinx = -1/4 + 3/4 or sinx = -1/4-3/4

sinx lies between -1 and 1 inclusive, so both roots are allowed

Thus the solutions are:
sinx = -1/4 + 3/4 = 1/2
or
sinx = -1/4-3/4 = -1

you can then sove each of these for x