Sin(2a) = Sin(a + a)

Sin(a + a) = Sin(a)Cos(a) + Cos(a)Sin(a) = 2Sin(a)Cos(a)

Cos(2a) = Cos(a +a)

Cos(a +a) = Cos(a)Cos(a) – Sin(a)Sin(a) =

By the Pythagorean theorem

and