Proving Trigonometry Identities: Help on a Practice Test.

Question

This was the first question: Let α=β=θ in the formula cos(α+β) and thus derive a formula for cos2θ

And I solved it as: cosθcosθ - sinθsinθ = cos² - sin²

Now, the question I can't figure out is: Use the identity sin² θ + cos²θ = 1 in order to write the above answer in terms of cosine only.

If anyone could help me out with that, I would be very thankful. Again, this is just the practice exam. The real exam is tomorrow.

Answer 1

If sin²(θ) + cos²(θ) = 1 then sin²(θ) = 1 - cos²(θ)

Substitute "1 - cos²(θ)" for "sin²(θ)" in cos(2θ) = cos²(θ) - sin²(θ) to get:

cos(2θ) = cos²(θ) - (1 - cos²(θ)) = 2cos²(θ) - 1