using the identities sin^2x+cos^2x=1, sin(x+y)=sinxcosy+cosxsiny, cos(x+y)=cosxcosy-sinxsiny prove the following 2 identities sin2theta=2sinthetacostheta and cos2theta=2cos^2theta-1

Identities:

sin^2(x) + cos^2(x) = 1

sin (x + y) = sin(x)cos(y) + cos(x)sin(y)

cos (x + y) = cos(x)cos(y) - sin(x)sin(y)

Prove:

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

cos(2x) = 2cos^2(x) - 1

The trick is to make x and y equal in the second two identities, and then to use the first identity to reduce.