Write each trigonometric expression as an algebraic expression in u, for u > 0.
sec (arccot (√4―u² )/ u)

Verified step by step guidance

1. Recognize that \( \text{arccot}(x) \) is the angle whose cotangent is \( x \). Therefore, if \( x = \frac{\sqrt{4-u^2}}{u} \), then \( \cot(\theta) = \frac{\sqrt{4-u^2}}{u} \).
2. Use the identity \( \cot(\theta) = \frac{1}{\tan(\theta)} \) to express \( \tan(\theta) \) in terms of \( u \): \( \tan(\theta) = \frac{u}{\sqrt{4-u^2}} \).
3. Recall the Pythagorean identity \( \sec^2(\theta) = 1 + \tan^2(\theta) \). Substitute \( \tan(\theta) = \frac{u}{\sqrt{4-u^2}} \) into this identity.
4. Calculate \( \tan^2(\theta) = \left(\frac{u}{\sqrt{4-u^2}}\right)^2 = \frac{u^2}{4-u^2} \).
5. Substitute \( \tan^2(\theta) \) into the Pythagorean identity to find \( \sec^2(\theta) = 1 + \frac{u^2}{4-u^2} \), and then take the square root to find \( \sec(\theta) \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as secant, sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The secant function, specifically, is defined as the reciprocal of the cosine function. Understanding these functions is essential for converting trigonometric expressions into algebraic forms.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arccotangent, are used to find angles when given a ratio. The arccotangent function returns the angle whose cotangent is the given value. This concept is crucial for interpreting expressions involving inverse functions and converting them into algebraic expressions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In the context of trigonometric expressions, this includes substituting trigonometric identities and simplifying ratios. Mastery of algebraic manipulation is necessary to express trigonometric functions in terms of a single variable, such as u.