What is the derivation of the power reducing formula for cosine?

Question

What is the proof for power reducing formula for cosine?

Answer 1

The power reducing formula for cosine is cos²Θ = 1/2(1 + cos2Θ)

Start with the double angle identity: cos2Θ = cos² Θ - sin²Θ

Now solve for cos²Θ by adding sin²Θ to both sides: cos2Θ + sin²Θ = cos²Θ

Use the reflexive property to write: cos²Θ = cos2Θ + sin²Θ

Since sin²Θ + cos²Θ = 1 replace sin²Θ with 1 - cos²Θ: cos²Θ = cos2Θ + 1 - cos²Θ

Add cos²Θ to both sides of the equation: 2cos²Θ = cos2Θ + 1

Multiply both sides by 2 to find cos²Θ: cos²Θ = 1/2(cos2Θ + 1)

Using the commutative property we can write: cos²Θ = 1/2(1 + cos2Θ)