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

Verified step by step guidance

1. Start by examining the left side of the equation: \(2 \cos^2 \left(\frac{x}{2}\right) \tan x\).
2. Use the double angle identity for cosine: \(\cos^2 \left(\frac{x}{2}\right) = \frac{1 + \cos x}{2}\). Substitute this into the equation.
3. The left side becomes: \(2 \left(\frac{1 + \cos x}{2}\right) \tan x = (1 + \cos x) \tan x\).
4. Now, expand the expression: \((1 + \cos x) \tan x = \tan x + \cos x \tan x\).
5. Recognize that \(\cos x \tan x = \sin x\), so the expression simplifies to \(\tan x + \sin x\), 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. Common identities include the Pythagorean identities, reciprocal identities, and co-function 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 cosine double angle formula states that cos(2θ) = cos²(θ) - sin²(θ). These formulas are often used to simplify expressions and verify identities involving angles that are multiples of a given angle.

Tangent Function and Its Relationship

The tangent function, defined as tan(x) = sin(x)/cos(x), relates the sine and cosine functions. Understanding this relationship is essential for manipulating and verifying equations involving tangent. Additionally, recognizing how tangent behaves in terms of periodicity and asymptotes can aid in solving trigonometric identities.