Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = 1 - 2 sin² x

Verified step by step guidance

1. Start with the left side of the equation: \( \cos 2x \).
2. Use the hint provided: express \( \cos 2x \) as \( \cos(x + x) \).
3. Apply the cosine addition formula: \( \cos(x + x) = \cos x \cos x - \sin x \sin x \).
4. Simplify the expression: \( \cos^2 x - \sin^2 x \).
5. Use the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \) to transform \( \cos^2 x - \sin^2 x \) into \( 1 - 2\sin^2 x \), which matches the right side of the equation.

Key Concepts

Trigonometric Identities

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

Double Angle Formulas

Double angle formulas express trigonometric functions of double angles in terms of single angles. For example, the cosine double angle formula states that cos(2x) can be expressed as cos²(x) - sin²(x) or 1 - 2sin²(x). These formulas are crucial for transforming and simplifying trigonometric expressions, particularly when verifying identities.

Verification of Identities

Verifying trigonometric identities involves showing that two sides of an equation are equivalent by manipulating one side using known identities and algebraic techniques. This process often requires strategic substitutions and simplifications, making it essential to have a strong grasp of various trigonometric identities and their relationships.