Verify that each equation is an identity.
(2 tan B)/(sin 2B) = sec² B

Verified step by step guidance

1. Start by recalling the double-angle identity for sine: \(\sin 2B = 2 \sin B \cos B\).
2. Rewrite the left-hand side (LHS) of the equation \(\frac{2 \tan B}{\sin 2B}\) by substituting \(\sin 2B\) with \(2 \sin B \cos B\), so it becomes \(\frac{2 \tan B}{2 \sin B \cos B}\).
3. Simplify the fraction by canceling the common factor 2 in numerator and denominator, resulting in \(\frac{\tan B}{\sin B \cos B}\).
4. Express \(\tan B\) in terms of sine and cosine: \(\tan B = \frac{\sin B}{\cos B}\), then substitute this into the expression to get \(\frac{\frac{\sin B}{\cos B}}{\sin B \cos B}\).
5. Simplify the complex fraction by multiplying numerator and denominator appropriately, which will lead to an expression involving \(\sec^2 B\), verifying the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means manipulating one side of the equation using known identities until it matches the other side, confirming their equivalence.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. For example, sin(2B) = 2 sin B cos B is essential for rewriting expressions involving sin 2B to simplify or transform the equation.

Basic Trigonometric Ratios and Reciprocal Identities

Understanding the definitions of tangent, secant, sine, and cosine, and their reciprocal relationships, such as sec B = 1/cos B and tan B = sin B/cos B, is crucial. These allow substitution and simplification to verify the given identity.