Verify that each equation is an identity.
(sin² θ)/cos θ = sec θ - cos θ

Verified step by step guidance

1. Start by rewriting the right-hand side of the equation using trigonometric identities: \( \sec \theta = \frac{1}{\cos \theta} \). So, the right-hand side becomes \( \frac{1}{\cos \theta} - \cos \theta \).
2. Combine the terms on the right-hand side over a common denominator: \( \frac{1}{\cos \theta} - \cos \theta = \frac{1 - \cos^2 \theta}{\cos \theta} \).
3. Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, \( 1 - \cos^2 \theta = \sin^2 \theta \).
4. Substitute \( \sin^2 \theta \) for \( 1 - \cos^2 \theta \) in the expression: \( \frac{\sin^2 \theta}{\cos \theta} \).
5. Observe that the left-hand side of the original equation is \( \frac{\sin^2 \theta}{\cos \theta} \), which matches the transformed right-hand side, thus verifying the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides of the equation are defined. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.

Reciprocal Functions

Reciprocal functions in trigonometry relate the sine, cosine, and tangent functions to their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). For example, sec θ is defined as 1/cos θ. Recognizing these relationships is essential for manipulating and verifying trigonometric equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations using algebraic rules. This includes factoring, combining like terms, and applying identities to transform one side of an equation to match the other. Mastery of these techniques is necessary for verifying trigonometric identities effectively.