Prove the double angle formula for cosine.

The double angle formula for cosine is cos(2θ) = cos²(θ) - sin²(θ).

To prove the double angle formula for cosine, we start with the addition formula for cosine:

cos(α + β) = cos(α)cos(β) - sin(α)sin(β)

Setting α = β = θ, we get:

cos(2θ) = cos(θ + θ) = cos(θ)cos(θ) - sin(θ)sin(θ)

Using the identity cos²(θ) + sin²(θ) = 1, we can substitute cos²(θ) = 1 - sin²(θ) to get:

cos(2θ) = (1 - sin²(θ)) - sin²(θ) = 1 - 2sin²(θ)

Rearranging, we get:

cos(2θ) = cos²(θ) - sin²(θ)

Therefore, we have proven the double angle formula for cosine.