1. The problem is to find an expression for $\cos 2a$ in terms of $\cos a$ and $\sin a$. 2. The double-angle formula for cosine is given by: $$\cos 2a = \cos^2 a - \sin^2 a$$ 3. Using the Pythagorean identity $\sin^2 a = 1 - \cos^2 a$, we can rewrite the formula as: $$\cos 2a = \cos^2 a - (1 - \cos^2 a) = 2\cos^2 a - 1$$ 4. Alternatively, using $\cos^2 a = 1 - \sin^2 a$, we get: $$\cos 2a = (1 - \sin^2 a) - \sin^2 a = 1 - 2\sin^2 a$$ 5. Therefore, the double-angle formula for cosine can be expressed in three equivalent ways: - $$\cos 2a = \cos^2 a - \sin^2 a$$ - $$\cos 2a = 2\cos^2 a - 1$$ - $$\cos 2a = 1 - 2\sin^2 a$$ These formulas are useful depending on which trigonometric function you know or want to use.