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Proving Trig Identities

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?

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