Write each trigonometric expression as an algebraic expression in u, for u > 0.
tan (sin⁻¹ u/(√u² + 2))

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Recognize that the expression is \( \tan \left( \sin^{-1} \left( \frac{u}{\sqrt{u^2 + 2}} \right) \right) \). Let \( \theta = \sin^{-1} \left( \frac{u}{\sqrt{u^2 + 2}} \right) \), so \( \sin \theta = \frac{u}{\sqrt{u^2 + 2}} \).

Recall the Pythagorean identity for sine and cosine: \( \sin^2 \theta + \cos^2 \theta = 1 \). Use this to find \( \cos \theta \) in terms of \( u \).

Calculate \( \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left( \frac{u}{\sqrt{u^2 + 2}} \right)^2} \). Simplify the expression inside the square root.

Use the definition of tangent in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute the expressions for \( \sin \theta \) and \( \cos \theta \) found in previous steps.

Simplify the resulting algebraic expression to write \( \tan \left( \sin^{-1} \left( \frac{u}{\sqrt{u^2 + 2}} \right) \right) \) purely in terms of \( u \).

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