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

See below.

Step-by-step 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.