Verify that each equation is an identity.
cos x = (1 - tan² (x/2))/(1 + tan² (x/2))

Verified step by step guidance

1. Recall the half-angle identity for tangent: \(\tan\left(\frac{x}{2}\right) = t\). We want to express \(\cos x\) in terms of \(t\).
2. Use the double-angle identity for cosine in terms of tangent of half-angle: \(\cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}\).
3. Substitute \(t = \tan\left(\frac{x}{2}\right)\) into the right-hand side to get \(\frac{1 - t^2}{1 + t^2}\).
4. Recognize that this expression matches the given right-hand side of the equation, so the identity holds if both sides are equal for all \(x\) where defined.
5. To verify, you can start from the left-hand side \(\cos x\) and rewrite it using the half-angle tangent substitution, or start from the right-hand side and simplify to \(\cos x\) using Pythagorean identities.

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, often using known formulas or transformations.

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, tan(x/2) can be related to sin x and cos x, enabling the rewriting of expressions like cos x in terms of tan²(x/2).

Algebraic Manipulation of Trigonometric Expressions

Algebraic manipulation involves rewriting and simplifying trigonometric expressions using identities, factoring, and common denominators. This skill is essential to transform one side of an equation to match the other when verifying identities.