Question 890642
you want to prove the identity:


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


your basic identity for cos(x + y) is:


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


let y = x, and this basic identity becomes:


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


simplify to get:


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


another basic identity is sin^2(x) + cos^2(x) = 1


from this basic identity, we can solve for cos^2(x) to get:


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


replace cos^2(x) with 1 - sin^2(x) in your equation of:


cos(2x) = cos^2(x) - sin^2(x) to get:


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


simplify this to get:


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


since that is equal to your original equation, you are done.