Simplify trigonometric expressions

The formula for simplifying \( \cos^3(x) \) is \( \cos^3(x) = \frac{1}{4}(3\cos(x) + \cos(3x)) \) using the triple angle formula.

The identity for \( \tan^2(x) \) is \( \tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)} \).

The Pythagorean identity is \( \sin^2(x) + \cos^2(x) = 1 \).

\( \sin^2(x) + \cos^2(x) = 1 \) based on the Pythagorean identity.

The double angle formula for sine is \( \sin(2x) = 2\sin(x)\cos(x) \).

The double angle formula for cosine is \( \cos(2x) = \cos^2(x) - \sin^2(x) \) or \( \cos(2x) = 2\cos^2(x) - 1 \) or \( \cos(2x) = 1 - 2\sin^2(x) \).

The relationship is \( \sin(x) = \cos(90° - x) \) and \( \cos(x) = \sin(90° - x) \).

The formula for \( \tan(x) \) is \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

Using the double angle formula, \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \).

The formula for simplifying \( \cos^2(x) \) is \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \).