Add x to both sides
x = 2x cos(x)

Divide both sides by 2x
cos(x) = 1/2

Take the arccosine of both sides
x = arccos(1/2)

1/2 is the cosine of π/3. Therefore, x = π/3.
(Technically, 7π/3, 13π/3, and -5π/3 are also valid answers, but they are all just ±2π representations of the same angle.)

The solution, for your convenience.

You fool! You just destroyed the universe!