K, so my teacher presented a set of problems and it is basically impossible to get all of them right (it was a competition to see who could do the most). There were a ton of problems and after going to class I still don't understand how to do some of them.

Disprove the following identity by counterexample:

#4

sin^2(x)+cos(x)=0

Everything that I plug in says that this isn't an identity. What am I doing wrong?

Prove the following identities:

#13

cotx-sinx=(cos^2(x)+cosx-1)csc(x)

I spent 2 hours trying to prove this, and I am coming with a nothing.

And the third one

Solve the following:

#14

1-2sin^2(2x)=3sin2x

I have a bunch of these that I didn't get right that I don't understand, but these 3 are really bugging me. I already know the answers, so I mostly care about how to get that answer because so far I haven't been able to figure it out. Any help will be great!!!