How would I prove an Identity like the cos(2x)=1-2(sinx)^2 or something like sin(2x)=2sinx*cosx. I don't really know how to get started

To prove sin(2x)=2sin(x)cos(x) use the addition formula for sine.

sin(u+v)=sin(u)cos(v)+cos(u)sin(v)

Then, we can write sin(2x)=sin(x+x)

and get:

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

See there?

To prove cos(2x)=1-2sin^{2}(x)

cos(2x)=cos(x+x)=\underbrace{cos(x)cos(x)-sin(x)sin(x)}_{\text{addition formula \\for cosine}}

cos^{2}(x)-sin^{2}(x)

\underbrace{(1-sin^{2}(x))}_{\text{cos^2(x)}}-sin^{2}(x)=1-2sin^{2}(x)

Easier than you thought, huh?