Answer:
- cos(2Ф) = cos²(Ф) -sin²(Ф)
- cos(2Ф) = 1 -2sin²(Ф)
- cos(2Ф) = 2cos²(Ф) -1
Step-by-step 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
