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 the expression is \( \sec(\arccot(\frac{\sqrt{4 - u^2}}{u})) \). Let \( \theta = \arccot\left(\frac{\sqrt{4 - u^2}}{u}\right) \). This means \( \cot \theta = \frac{\sqrt{4 - u^2}}{u} \).
2. Recall the definition of cotangent: \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \). So, we can think of a right triangle where the adjacent side is \( \sqrt{4 - u^2} \) and the opposite side is \( u \).
3. Find the hypotenuse of this right triangle using the Pythagorean theorem: \( \text{hypotenuse} = \sqrt{(\sqrt{4 - u^2})^2 + u^2} = \sqrt{4 - u^2 + u^2} = \sqrt{4} = 2 \).
4. Recall that \( \sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} \). Using the triangle sides, \( \sec \theta = \frac{2}{\sqrt{4 - u^2}} \).
5. Therefore, the original expression \( \sec(\arccot(\frac{\sqrt{4 - u^2}}{u})) \) can be written algebraically as \( \frac{2}{\sqrt{4 - u^2}} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like arccot, return an angle whose trigonometric ratio matches the given value. Understanding how to interpret arccot(x) as an angle θ such that cot(θ) = x is essential for rewriting expressions involving inverse functions.

Trigonometric Ratios and Identities

Trigonometric ratios (sine, cosine, tangent, secant, etc.) relate the sides of a right triangle to its angles. Knowing identities such as sec(θ) = 1/cos(θ) and cot(θ) = adjacent/opposite helps convert trigonometric expressions into algebraic forms.

Right Triangle Representation and Algebraic Substitution

Representing the angle from an inverse trig function as a right triangle allows expressing trigonometric functions in terms of side lengths. Using the given expression inside arccot, one can assign sides and use the Pythagorean theorem to rewrite sec(arccot(...)) as an algebraic expression in u.