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

Verified step by step guidance

1. Start by expressing \( \sec \alpha \) and \( \tan \alpha \) in terms of sine and cosine: \( \sec \alpha = \frac{1}{\cos \alpha} \) and \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \).
2. Substitute these expressions into the left-hand side of the equation: \( \frac{1}{\sec \alpha - \tan \alpha} = \frac{1}{\frac{1}{\cos \alpha} - \frac{\sin \alpha}{\cos \alpha}} \).
3. Simplify the denominator by combining the fractions: \( \frac{1}{\frac{1 - \sin \alpha}{\cos \alpha}} = \frac{\cos \alpha}{1 - \sin \alpha} \).
4. Now, consider the right-hand side: \( \sec \alpha + \tan \alpha = \frac{1}{\cos \alpha} + \frac{\sin \alpha}{\cos \alpha} = \frac{1 + \sin \alpha}{\cos \alpha} \).
5. Multiply the numerator and the denominator of the left-hand side by the conjugate of the denominator \( 1 + \sin \alpha \) to simplify and verify that both sides are equal: \( \frac{\cos \alpha (1 + \sin \alpha)}{(1 - \sin \alpha)(1 + \sin \alpha)} = \frac{1 + \sin \alpha}{\cos \alpha} \).

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 and simplifying expressions in trigonometry.

Reciprocal Functions

Reciprocal functions in trigonometry include secant (sec) and cosecant (csc), which are the reciprocals of cosine and sine, respectively. The secant function is defined as sec α = 1/cos α, and the tangent function is defined as tan α = sin α/cos α. Recognizing these relationships is essential for manipulating and verifying trigonometric equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, expanding, and combining like terms. In the context of trigonometric identities, effective algebraic manipulation allows one to transform one side of an equation to match the other, thereby verifying the identity.