Use an angle sum identity to verify the identity cos 2 0=2 cos 0-1
To verify the identity cos 2θ = 2 cos θ - 1 using an angle sum identity, we need to use the double-angle formula for cosine.
The double-angle formula for cosine states that cos 2θ = cos²θ - sin²θ.
First, let's rewrite the right side of the identity, 2 cos θ - 1, using the double-angle formula:
2 cos θ - 1 = 2(cos²θ - sin²θ) - 1
= 2cos²θ - 2sin²θ - 1
Now, we need to express sin²θ in terms of cosθ using another trigonometric identity. The Pythagorean identity, sin²θ + cos²θ = 1, can be rearranged as sin²θ = 1 - cos²θ.
Substituting sin²θ = 1 - cos²θ into our expression:
2cos²θ - 2(1 - cos²θ) - 1
= 2cos²θ - 2 + 2 cos²θ - 1
= 4cos²θ - 3
Now, we can check if this expression is equal to the left side of the identity, cos 2θ:
cos 2θ = 4cos²θ - 3
Since the expression on the right side is equal to the left side, we have verified the identity cos 2θ = 2cosθ - 1 using the double-angle formula for cosine.