What does the cosine of a double angle, cos(2x), equal?

The cosine of a double angle, denoted as cos(2x), is a well-known identity in trigonometry. The correct identity states that cos(2x) can be expressed as 2cos²(x) - 1. This formulation comes from the angle addition formulas and can be derived from the identity relating cosine to sine.

When considering the double angle formula, it helps to start with the addition formula for cosine:

[ \cos(2x) = \cos(x + x) = \cos(x) \cos(x) - \sin(x) \sin(x). ]

This can be simplified as follows:

[ \cos(2x) = \cos^2(x) - \sin^2(x). ]

Using the Pythagorean identity, we know that (\sin^2(x) = 1 - \cos^2(x)), substituting this into the equation gives:

[ \cos(2x) = \cos^2(x) - (1 - \cos^2(x)) = \cos^2(x) - 1 + \cos^2(x) = 2\cos^2(x) - 1. ]

This shows that the expression