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. Substitute \( \sin 2B \) in the equation with \( 2 \sin B \cos B \), so the left side becomes \( \frac{2 \tan B}{2 \sin B \cos B} \).
3. Simplify the left side: \( \frac{2 \tan B}{2 \sin B \cos B} = \frac{\tan B}{\sin B \cos B} \).
4. Recall that \( \tan B = \frac{\sin B}{\cos B} \), substitute this into the equation: \( \frac{\frac{\sin B}{\cos B}}{\sin B \cos B} \).
5. Simplify the expression: \( \frac{\sin B}{\cos B} \times \frac{1}{\sin B \cos B} = \frac{1}{\cos^2 B} = \sec^2 B \), 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. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.

Double Angle Formulas

Double angle formulas express trigonometric functions of double angles in terms of single angles. For example, the sine double angle formula states that sin(2B) = 2sin(B)cos(B). These formulas are essential for transforming expressions and simplifying the verification of identities involving angles that are doubled.

Secant Function

The secant function, denoted as sec(B), is the reciprocal of the cosine function, defined as sec(B) = 1/cos(B). It plays a significant role in trigonometric identities and equations. Recognizing how secant relates to other trigonometric functions is vital for manipulating and verifying equations involving sec² B.