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

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Recognize that the expression involves the inverse secant function: \( \sec^{-1}\left(\frac{u}{2}\right) \). Let \( \theta = \sec^{-1}\left(\frac{u}{2}\right) \), so that \( \sec \theta = \frac{u}{2} \).

Recall the identity for sine of a double angle: \( \sin(2\theta) = 2 \sin \theta \cos \theta \). Our goal is to express \( \sin(2 \sec^{-1}(u/2)) = \sin(2\theta) \) in terms of \( u \).

Since \( \sec \theta = \frac{1}{\cos \theta} = \frac{u}{2} \), solve for \( \cos \theta \): \( \cos \theta = \frac{2}{u} \).

Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \): \( \sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{2}{u}\right)^2} = \sqrt{1 - \frac{4}{u^2}} \). Since \( u > 0 \), take the positive root.

Substitute \( \sin \theta \) and \( \cos \theta \) into the double angle formula: \( \sin(2\theta) = 2 \sin \theta \cos \theta = 2 \times \sqrt{1 - \frac{4}{u^2}} \times \frac{2}{u} \). Simplify this expression to write \( \sin(2 \sec^{-1}(u/2)) \) purely in terms of \( u \).

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Inverse Trigonometric Functions

Inverse trigonometric functions, like sec⁻¹(u/2), return an angle whose trigonometric ratio equals the given value. Understanding how to interpret and manipulate these functions is essential to rewrite expressions involving angles defined by inverse functions.

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Introduction to Inverse Trig Functions

Double-Angle Formulas

Double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. For sine, sin(2θ) = 2 sin θ cos θ, which helps convert sin(2 sec⁻¹(u/2)) into an expression involving sin and cos of sec⁻¹(u/2).

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Double Angle Identities

Right Triangle and Algebraic Substitution

By interpreting sec⁻¹(u/2) as an angle in a right triangle with known side ratios, one can express sine and cosine in terms of u. This geometric approach allows converting trigonometric expressions into algebraic forms using Pythagorean identities.

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