Use sum identities to derive one double angle identity for cosine

Hint: cos 2 Ф = cos (Ф+Ф)

Ф=theta

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

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

cos (2 Ф) = (1 - sin² (Ф)) - sin² (Ф)

cos (2 Ф) = 1 - 2sin² (Ф)

or

cos (2 Ф) = cos² (Ф) - (1 - cos² (Ф))

cos (2 Ф) = 2cos² (Ф) - 1