sin 2x + 3 cos 2x = 0
This isn't an identity!
An identity is an equation, but it is one which is true for all values of x.
For example, sin^2(x) + cos^2(x) = 1 is an identity.
It doesn't matter what value you give to x, sin^2x) + cos^2(x) will always be equal to one.
Also, sin(2x) + 3.cos(2x) = 0 doesn't work for x = 0!
What you have is a trig equation, with x being the unknown.
Let us solve this equation for x
sin(2x) + 3.cos(2x) = 0
sin(2x) = -3.cos(2x)
sin(2x)/cos(2x) = -3
tan(2x) = -3
And from my calculator I get: 2x = -1.249 rads
Answer: x = -0.6245 rads