How to solve

3sin^2x= cos^2x

I'll try it.
I just went through this twice on scratch paper.  The first time was to
see if I could do it, and the second time was because the first result
I got was ridiculous.  But I think I got it.

You said                                                 3sin²(x) = cos²(x)

Use this trig identity:  sin²(x) = 1 - cos²(x)   

                                   Plug it into the original equation for (x).

                                                     3(1 - cos²(x) )  =  cos²(x)

Remove parentheses on the left:    3 - 3cos²(x)  =  cos²(x)

Add  3cos²(x)  to each side:            3                  =  4cos²(x)

Divide each side by  4 :                  3/4  =  cos²(x)

Take the square root of each side:    cos(x) = (√3) / 2 .

There it is ... the cosine of the unknown angle.
Now you just go look it up in a book with a table cosines,
or else pinch it through your computer or your calculator,
or else just remember that you've learned that

                           cos( 30° )  =  (√3) / 2 .