Verify that each equation is an identity.
(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)

Verified step by step guidance

1. Start by recognizing that the right side of the equation, \( \cos^2 \left( \frac{x}{2} \right) - \sin^2 \left( \frac{x}{2} \right) \), is a known trigonometric identity for \( \cos(x) \).
2. Rewrite the right side using the identity: \( \cos^2 \left( \frac{x}{2} \right) - \sin^2 \left( \frac{x}{2} \right) = \cos(x) \).
3. Now, focus on the left side: \( \frac{\sin(2x)}{2\sin(x)} \). Use the double angle identity for sine: \( \sin(2x) = 2\sin(x)\cos(x) \).
4. Substitute the double angle identity into the left side: \( \frac{2\sin(x)\cos(x)}{2\sin(x)} \).
5. Simplify the expression by canceling \( 2\sin(x) \) in the numerator and denominator, which leaves you with \( \cos(x) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.

Double Angle Formulas

Double angle formulas express trigonometric functions of double angles in terms of single angles. For example, sin(2x) = 2sin(x)cos(x) and cos(2x) = cos²(x) - sin²(x). These formulas are essential for transforming and simplifying expressions involving angles that are multiples of a given angle.

Verification of Identities

Verifying trigonometric identities involves manipulating one side of the equation to show that it is equivalent to the other side. This process often requires the use of algebraic techniques and known identities. It is a fundamental skill in trigonometry that helps in understanding the relationships between different trigonometric functions.