Verify that each equation is an identity.
2 cos² θ - 1 = (1 - tan² θ)/(1 + tan² θ)

Verified step by step guidance

1. Start by recalling the Pythagorean identity for cosine squared: \(\cos^{2} \theta = \frac{1}{1 + \tan^{2} \theta}\). This will help relate cosine and tangent functions.
2. Rewrite the left side of the equation, \(2 \cos^{2} \theta - 1\), by substituting \(\cos^{2} \theta\) with \(\frac{1}{1 + \tan^{2} \theta}\), so it becomes \(2 \times \frac{1}{1 + \tan^{2} \theta} - 1\).
3. Simplify the expression on the left side by combining the terms over a common denominator: \(\frac{2}{1 + \tan^{2} \theta} - 1 = \frac{2 - (1 + \tan^{2} \theta)}{1 + \tan^{2} \theta}\).
4. Simplify the numerator: \(2 - (1 + \tan^{2} \theta) = 1 - \tan^{2} \theta\), so the left side becomes \(\frac{1 - \tan^{2} \theta}{1 + \tan^{2} \theta}\).
5. Compare this simplified left side expression with the right side of the original equation, \(\frac{1 - \tan^{2} \theta}{1 + \tan^{2} \theta}\), and conclude that both sides are equal, verifying the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They are used to simplify expressions or prove equivalences, such as the Pythagorean identities and angle sum formulas.

Pythagorean Identities

Pythagorean identities relate the squares of sine, cosine, and tangent functions, such as sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ. These identities are fundamental for transforming and verifying trigonometric expressions.

Expressing Functions in Terms of Sine and Cosine

Tangent and other trigonometric functions can be expressed as ratios of sine and cosine (e.g., tan θ = sin θ / cos θ). Rewriting expressions this way helps in simplifying and verifying identities by using common functions.