Write each trigonometric expression as an algebraic expression in u, for u > 0.
tan (arcsec (√1―u²) / u)

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Recognize that the expression is \( \tan(\arcsec(\frac{\sqrt{1 - u^2}}{u})) \). Here, \( \arcsec(x) \) is the inverse secant function, which returns an angle \( \theta \) such that \( \sec(\theta) = x \).

Set \( \theta = \arcsec\left(\frac{\sqrt{1 - u^2}}{u}\right) \). By definition, this means \( \sec(\theta) = \frac{\sqrt{1 - u^2}}{u} \).

Recall the identity \( \sec(\theta) = \frac{1}{\cos(\theta)} \), so \( \cos(\theta) = \frac{u}{\sqrt{1 - u^2}} \).

Use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) to find \( \sin(\theta) \). Substitute \( \cos(\theta) \) and solve for \( \sin(\theta) \).

Finally, express \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) in terms of \( u \) using the expressions found for \( \sin(\theta) \) and \( \cos(\theta) \).

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