Question

Use sum identities to derive one double angle identity for cosine

Hint: cos 2 Ф = cos (Ф+Ф)
Ф=theta

Answer 1

• cos(2Ф) = cos²(Ф) -sin²(Ф)
• cos(2Ф) = 1 -2sin²(Ф)
• cos(2Ф) = 2cos²(Ф) -1

Explanation:

The angle sum formula for cosine is ...

cos(α+β) = cos(α)cos(β) -sin(α)sin(β)

When we have α = β = Ф, this becomes ...

cos(Ф+Ф) = cos(Ф)cos(Ф) -sin(Ф)sin(Ф)

cos(2Ф) = cos²(Ф) -sin²(Ф)

The "Pythagorean identity" can be used to write this in terms of sine or cosine.

cos(2Ф) = (1 -sin²(Ф)) -sin²(Ф)

cos(2Ф) = 1 -2sin²(Ф)

or

cos(2Ф) = cos²(Ф) -(1 -cos²(Ф))

cos(2Ф) = 2cos²(Ф) -1