Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α

Verified step by step guidance

1. Start by recalling the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \).
2. Recall the identity for tangent and secant: \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) and \( \sec^2 \alpha = 1 + \tan^2 \alpha \).
3. Substitute \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \) into the equation: \( \sin^2 \alpha + \frac{\sin^2 \alpha}{\cos^2 \alpha} + \cos^2 \alpha = \sec^2 \alpha \).
4. Combine the terms on the left side over a common denominator: \( \frac{\sin^2 \alpha \cdot \cos^2 \alpha + \sin^2 \alpha + \cos^4 \alpha}{\cos^2 \alpha} \).
5. Simplify the expression and verify if it equals \( \sec^2 \alpha \), using the identity \( \sec^2 \alpha = 1 + \tan^2 \alpha \).

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle α, the relationship sin² α + cos² α = 1 holds true. This fundamental identity is crucial in trigonometry as it connects the sine and cosine functions, allowing for simplifications and transformations in various equations.

Definition of Tangent and Secant

Tangent and secant are defined in terms of sine and cosine: tan α = sin α / cos α and sec α = 1 / cos α. Understanding these definitions is essential for manipulating trigonometric identities, as they allow us to express equations in terms of sine and cosine, facilitating verification 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 the Pythagorean identities, reciprocal identities, and quotient identities. Recognizing and applying these identities is key to verifying equations and simplifying trigonometric expressions.