Question

How to solve? cos( 2theta)+10 sin^2 theta =7

Answer 1

I'll switch to x instead of theta, typing is a lot easier then...
Your equation: cos2x+10sin²x=7.

There is a (well-known) formula for cos2x: cos2x=1-2sin²x.
If we put that in place of cos2x, we get:

1-2sin²x+10sin²x=7, which after cleaning up becomes:

8sin²x=6.

I think it's your turn to do the next step now...