Double angles (C)

It follows directly from the addition formulae that

$$\cos(2A) = \cos^2(A) - \sin^2(A)$$

and

$$\sin(2A) = 2\sin(A)\cos(A).$$

Exercises:

1.

Derive these two identities.

2.

Use the circle identity to show that

$$\cos(2A) = 2\cos^2(A) - 1$$

and

$$\cos(2A) = 1-2\sin^2(A).$$

3.

Derive identities for $\cos(3A)$ and $\sin(3A)$.