If sin θ = a and cos θ = b, represent each of the following in terms of a and b.tan θ - sec θ

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Start by recalling the definitions of tangent and secant in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).

Substitute the given values \( \sin \theta = a \) and \( \cos \theta = b \) into the expressions for tangent and secant.

For \( \tan \theta \), substitute to get \( \tan \theta = \frac{a}{b} \).

For \( \sec \theta \), substitute to get \( \sec \theta = \frac{1}{b} \).

Combine the expressions to find \( \tan \theta - \sec \theta = \frac{a}{b} - \frac{1}{b} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, which states that sin²θ + cos²θ = 1, and the definitions of tangent and secant in terms of sine and cosine. Understanding these identities is crucial for manipulating and simplifying expressions in trigonometry.

Tangent and Secant Functions

The tangent function, tan θ, is defined as the ratio of the sine and cosine functions: tan θ = sin θ / cos θ. The secant function, sec θ, is the reciprocal of the cosine function: sec θ = 1 / cos θ. Knowing how to express these functions in terms of sine and cosine allows for the conversion of expressions involving tan θ and sec θ into forms that can be expressed using a and b.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In trigonometry, this includes substituting known values, factoring, and combining like terms. Mastery of algebraic manipulation is essential for transforming trigonometric expressions into desired forms, such as expressing tan θ - sec θ in terms of a and b.

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