In Exercises 1–60, verify each identity.cos² θ (1 + tan² θ) = 1

Verified step by step guidance

1. Start by recalling the Pythagorean identity: \(1 + \tan^2 \theta = \sec^2 \theta\).
2. Substitute \(\sec^2 \theta\) for \(1 + \tan^2 \theta\) in the given expression: \(\cos^2 \theta (\sec^2 \theta)\).
3. Remember that \(\sec \theta = \frac{1}{\cos \theta}\), so \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\).
4. Substitute \(\frac{1}{\cos^2 \theta}\) for \(\sec^2 \theta\) in the expression: \(\cos^2 \theta \cdot \frac{1}{\cos^2 \theta}\).
5. Simplify the expression: \(\cos^2 \theta \cdot \frac{1}{\cos^2 \theta} = 1\), verifying the identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin² θ + cos² θ = 1. This fundamental relationship between sine and cosine is crucial for simplifying trigonometric expressions and verifying identities. It allows us to express one function in terms of another, facilitating the manipulation of equations.

Tangent and Secant Relationship

The tangent function is defined as tan θ = sin θ / cos θ, and it can also be expressed in terms of secant: tan² θ = sec² θ - 1. This relationship is essential for transforming expressions involving tangent into forms that can be more easily simplified or verified, particularly in the context of identities.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include reciprocal identities, quotient identities, and Pythagorean identities. Understanding these identities is vital for verifying trigonometric equations, as they provide the tools needed to manipulate and simplify expressions.