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

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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 \( \cot \theta = \frac{\sqrt{4 - u^2}}{u} \).

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 \( u \).

Find the hypotenuse of this right triangle using the Pythagorean theorem: \( \text{hypotenuse} = \sqrt{(\sqrt{4 - u^2})^2 + u^2} = \sqrt{4 - u^2 + u^2} = \sqrt{4} = 2 \).

Recall that \( \sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} \). Using the triangle sides, \( \sec \theta = \frac{2}{\sqrt{4 - u^2}} \).

Therefore, the original expression \( \sec(\arccot(\frac{\sqrt{4 - u^2}}{u})) \) can be written algebraically as \( \frac{2}{\sqrt{4 - u^2}} \).

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