Question

Prove 8cos^4(θ) = 3 + 4cos(2θ) + cos(4θ)

Answer 1

See below.

Explanation:

We make use of the double angle formula :

cos 2θ = 2 cos^2θ - 1.

8cos^4(θ) = 3 + 4cos(2θ) + cos(4θ)

8cos^4(θ) = 3 + 4(2cos^2θ - 1) + cos(4θ)

8cos^4(θ) = 3 + 8cos^2θ - 4 + cos(4θ)

8cos^4(θ) = 8cos^2 θ - 1 + cos(4θ)

8cos^4(θ) = 8cos^2 θ - 1 + 2cos^2 2θ - 1

8cos^4(θ) = 2(4cos^2 θ + cos^2 2θ - 1)

4cos^4(θ) = 4cos^2 θ + cos^2 2θ - 1

4cos^4(θ) = 4cos^2 θ + (2 cos^2 θ - 1)^2- 1

4cos^4(θ) = 4cos^2 θ + 4cos^4 θ - 4cos^2θ + 1 - 1

4 cos^4 θ = 4 cos^4 θ.

- Verified.

Phew!! That was a long one.