$\displaystyle cos^2(x)-cos2(x)+sin(x)=-sin^2(x)$

There's the problem, I've tried various different solutions but I can't seem to get it right, I've noticed that the middle cosine is a double-angle but I can't seem to make the formula give me an actual answer.

For example, after canceling out and using the double-angle formula of $\displaystyle cos2(x)=cos^2(x)-sin^2(x)$ I reached: $\displaystyle -sin^2(x)+sin(x)=-sin^2(x)$ So...can that even be solved like that or have I done something wrong?

$\displaystyle csc^2(x)-csc(x)=2$

Same as the one above, despite the fact that it looks simpler, I'm basically incompetent. I have a slew of other problems similar to these and I think some examples will really help me figure out the others. Maybe there's some basic formula/identity that I'm missing out on.

Note: I'm sorry for the title, I couldn't think of anything more specific. At least I'm following all the rules this time, right? (Happy)