Verify that each equation is an identity.
(sec α + csc α) (cos α - sin α) = cot α - tan α

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Start by expressing all trigonometric functions in terms of sine and cosine: \( \sec \alpha = \frac{1}{\cos \alpha} \), \( \csc \alpha = \frac{1}{\sin \alpha} \), \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \), and \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \).

Rewrite the left side of the equation: \((\sec \alpha + \csc \alpha)(\cos \alpha - \sin \alpha) = \left(\frac{1}{\cos \alpha} + \frac{1}{\sin \alpha}\right)(\cos \alpha - \sin \alpha)\).

Simplify the expression: Combine the terms inside the parentheses on the left side to get a common denominator: \(\frac{\sin \alpha + \cos \alpha}{\sin \alpha \cos \alpha}\).

Multiply the simplified expression by \((\cos \alpha - \sin \alpha)\): \(\frac{(\sin \alpha + \cos \alpha)(\cos \alpha - \sin \alpha)}{\sin \alpha \cos \alpha}\).

Simplify the expression further by expanding the numerator and comparing it to the right side of the equation: \(\cot \alpha - \tan \alpha = \frac{\cos^2 \alpha - \sin^2 \alpha}{\sin \alpha \cos \alpha}\).

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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.

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Fundamental Trigonometric Identities

Reciprocal Functions

Reciprocal functions in trigonometry include secant (sec), cosecant (csc), cotangent (cot), and tangent (tan). These functions are defined as the reciprocals of the basic trigonometric functions: sec α = 1/cos α, csc α = 1/sin α, cot α = 1/tan α, and tan α = sin α/cos α. Recognizing these relationships is essential for manipulating and verifying trigonometric equations.

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Secant, Cosecant, & Cotangent on the Unit Circle

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In trigonometry, this includes factoring, distributing, and combining like terms. Mastery of these techniques is necessary to transform one side of an equation into the other, which is a key step in verifying trigonometric identities.

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