Verify that each equation is an identity.
(sec α - tan α)² = (1 - sin α)/(1 + sin α)

Verified step by step guidance

1. Start by expanding the left side of the equation: \((\sec \alpha - \tan \alpha)^2\).
2. Recall the identities: \(\sec \alpha = \frac{1}{\cos \alpha}\) and \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\).
3. Substitute these identities into the expression: \((\frac{1}{\cos \alpha} - \frac{\sin \alpha}{\cos \alpha})^2\).
4. Simplify the expression: \((\frac{1 - \sin \alpha}{\cos \alpha})^2\).
5. Square the expression: \(\frac{(1 - \sin \alpha)^2}{\cos^2 \alpha}\) and verify it equals \(\frac{1 - \sin \alpha}{1 + \sin \alpha}\) by simplifying both sides.

Key Concepts

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. Understanding these identities is crucial for verifying equations, as they allow for the transformation of one side of the equation into the other.

Secant and Tangent Functions

The secant function, sec(α), is defined as the reciprocal of the cosine function, while the tangent function, tan(α), is the ratio of the sine function to the cosine function. These functions are essential in trigonometry and often appear in identities and equations. Recognizing how to manipulate these functions is key to simplifying and verifying trigonometric expressions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, expanding, and combining like terms. In the context of verifying trigonometric identities, effective algebraic manipulation allows one to transform one side of the equation into the other, demonstrating that both sides are equivalent.