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 writing down the given equation to verify: \(2 \cos^{2}\left(\frac{x}{2}\right) \tan x = \tan x + \sin x\).
2. Recall the double-angle identity for cosine: \(\cos x = 2 \cos^{2}\left(\frac{x}{2}\right) - 1\). From this, express \(2 \cos^{2}\left(\frac{x}{2}\right)\) as \(1 + \cos x\).
3. Substitute \(2 \cos^{2}\left(\frac{x}{2}\right)\) with \(1 + \cos x\) in the left side of the equation, so it becomes \((1 + \cos x) \tan x\).
4. Rewrite \(\tan x\) as \(\frac{\sin x}{\cos x}\) and distribute on the left side: \((1 + \cos x) \cdot \frac{\sin x}{\cos x} = \frac{\sin x}{\cos x} + \frac{\sin x \cos x}{\cos x}\).
5. Simplify the right side of the expression by canceling \(\cos x\) in the second term, resulting in \(\frac{\sin x}{\cos x} + \sin x\), which matches the right side of the original equation, confirming 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 showing both sides simplify to the same expression using known formulas, such as Pythagorean or angle-sum identities.

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, cos²(x/2) can be rewritten using the identity cos²(θ) = (1 + cos(2θ))/2, which helps simplify expressions involving cos²(x/2).

Relationship Between Tangent, Sine, and Cosine

Tangent is defined as sin(x)/cos(x). Understanding this relationship allows rewriting tan x in terms of sine and cosine, facilitating simplification and comparison of both sides of the equation when verifying identities.