Using the equation cos(a+b) = cos(a)cos(b) - sin(a)sin(b) or otherwise, show that cos(2x) = 2cos^2(x) - 1.

First let a = b = x such that:          

          cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

becomes:

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

Leading to:

          cos(2x) = cos²(x) - sin²(x)

Using the fact that sin²(y) + cos²(y) = 1 or rearranged sin²(y) = 1 - cos²(y):

          cos(2x) = cos²(x) - (1 - cos²(y)) = 2cos²(x) - 1, as required.

Another suitable approach may involve the Maclaurin series of cos(2x) and cos²(x) to arrive at the required relation, although this is more involved.