Evaluate each expression without using a calculator.
sec (sec⁻¹ 2)

Verified step by step guidance

1. Recognize that the expression is \( \sec(\sec^{-1} 2) \). Here, \( \sec^{-1} 2 \) is the inverse secant function, which gives an angle \( \theta \) such that \( \sec \theta = 2 \).
2. Let \( \theta = \sec^{-1} 2 \). By definition, this means \( \sec \theta = 2 \).
3. The original expression \( \sec(\sec^{-1} 2) \) can be rewritten as \( \sec(\theta) \). Since \( \theta \) was chosen so that \( \sec \theta = 2 \), substitute this value back.
4. Therefore, \( \sec(\sec^{-1} 2) = \sec \theta = 2 \).
5. This shows that applying \( \sec \) to its inverse function \( \sec^{-1} \) returns the original input value, as long as it is within the domain of the inverse function.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like sec⁻¹(x), return the angle whose trigonometric function equals x. For sec⁻¹ 2, it means finding the angle θ such that sec(θ) = 2. Understanding the domain and range of these inverse functions is essential for correct evaluation.

Secant Function (sec θ)

The secant function is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Knowing this relationship helps in simplifying expressions and understanding how sec and its inverse interact, especially when evaluating compositions like sec(sec⁻¹ x).

Composition of Functions and Simplification

When a function is composed with its inverse, such as sec(sec⁻¹ x), the result is typically x within the domain of the inverse function. Recognizing this property allows for direct simplification without calculation, provided the input lies within the valid range.