Verify that each equation is an identity.
(sin 2x)/(sin x) = 2/sec x

Verified step by step guidance

1. Start by rewriting the left side of the equation: \( \frac{\sin 2x}{\sin x} \). Use the double angle identity for sine: \( \sin 2x = 2 \sin x \cos x \).
2. Substitute the double angle identity into the equation: \( \frac{2 \sin x \cos x}{\sin x} \).
3. Simplify the expression by canceling \( \sin x \) in the numerator and the denominator: \( 2 \cos x \).
4. Now, consider the right side of the equation: \( \frac{2}{\sec x} \). Recall that \( \sec x = \frac{1}{\cos x} \), so \( \frac{2}{\sec x} = 2 \cos x \).
5. Compare both sides: After simplification, both sides of the equation are \( 2 \cos x \), thus 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. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.

Sine and Secant Functions

The sine function, denoted as sin(x), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. The secant function, sec(x), is the reciprocal of the cosine function, defined as 1/cos(x). Familiarity with these functions and their relationships is essential for manipulating and verifying trigonometric equations.

Algebraic Manipulation in Trigonometry

Algebraic manipulation involves rearranging and simplifying equations using algebraic techniques. In trigonometry, this includes factoring, finding a common denominator, and applying identities to transform one side of an equation to match the other. Mastery of these skills is necessary for effectively verifying trigonometric identities.