Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = cos² x - sin² x

Verified step by step guidance

1. Start by using the hint provided: express \( \cos 2x \) as \( \cos(x + x) \).
2. Apply the angle addition formula for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
3. Substitute \( a = x \) and \( b = x \) into the formula: \( \cos(x + x) = \cos x \cos x - \sin x \sin x \).
4. Simplify the expression: \( \cos x \cos x = \cos^2 x \) and \( \sin x \sin x = \sin^2 x \).
5. Conclude that \( \cos 2x = \cos^2 x - \sin^2 x \), verifying the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides of the equation are defined. They are fundamental in simplifying expressions and solving trigonometric equations. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas.

Double Angle Formulas

Double angle formulas express trigonometric functions of double angles in terms of single angles. For cosine, the double angle formula is cos(2x) = cos²(x) - sin²(x). This formula is essential for transforming expressions involving double angles into simpler forms, facilitating verification of identities.

Cosine Function Properties

The cosine function is a periodic function that represents the x-coordinate of a point on the unit circle corresponding to a given angle. Its properties include evenness (cos(-x) = cos(x)) and periodicity (cos(x + 2π) = cos(x)). Understanding these properties is crucial for manipulating and verifying trigonometric identities.