Answer to Question #302224 in Trigonometry for salomo

Question

Qn 4. For a given angle θ, suppose that

sin2 θ + cos2 θ = 1 and cos2 θ − sin2 θ = cos 2θ.

2Derive an expression for cos 4θ in terms only of powers of cos θ

Answer 1

$$\cos 2θ=\cos ^2θ-\sin ^2θ=2\cos^2\theta-1$$

$$\cos 4θ=2\cos^22\theta-1=2(2\cos^2\theta-1)^2-1$$

$$=8\cos^4\theta-8\cos^2\theta+1$$