trigonometric equation solve

Are solving & proving the same thing? My book has a method for "proving" using either the quotient identity or pythagorean identity - you know that whole, LS=RS stuff.

Earlier, though there is a method for "solving" by "let k=cos x" for example.

Well, for some of my assignment, I used the "solving" method *because the question told me to solve* and when it was returned to me, the teacher's notes were:

your factoring is rushed and solving is based on two identities: the tangent and sin^2x+cos^2x=1. You will be able to solve all by using these two.

I don't know what is going on.

If anyone can point me to a nice basic thread on MHP/website on this stuff I will be ever-so grateful!