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The double-angle formula is a crucial trigonometric identity that simplifies expressions involving angles that are twice a given angle. These formulas are used often in trigonometry because they help to work more efficiently with angles. In this problem, we're specifically interested in the sine double-angle formula,

• \( \sin 2\theta = 2 \sin \theta \cos \theta \).

This formula allows us to express the sine of a double angle directly in terms of the sine and cosine of the original angle. It's particularly useful because it expands our toolkit for transforming and simplifying trigonometric equations.
When we set out to show that \( f(\tan \theta) = \sin 2\theta \), the double-angle formula becomes the bridge from a traditional trigonometric form to the desired outcome. In practice, once you've simplified your complex expression—usually involving tangents or other trigonometric functions—the double-angle identity helps you to arrive at a neat solution, often transforming unwieldy terms into elegant summaries that reveal underlying equivalences.

The Pythagorean identity is one of the cornerstone identities in trigonometry. It comes from the Pythagorean theorem and applies to sine and cosine for angles in trigonometric functions. The most common form of this identity is:

• \( \cos^2 \theta + \sin^2 \theta = 1 \).

This identity is indispensable for transforming expressions into different forms by using the inherent relationship between cosine and sine.
In this exercise, the identity aids in breaking down \( (\tan \theta)^2 \). By using the relationship \( (\tan \theta)^2 = \frac{\sin^2 \theta}{\cos^2 \theta} \), we can rewrite parts of our expression involving tangents into more workable terms, explicitly linking back to the fundamental trigonometric functions. This transformation is critical as it allows the original function from the problem to be rewritten and eventually solved using other trigonometric identities like the double-angle formula.

Function transformation is a powerful tool when solving trigonometric problems. It allows you to modify and simplify expressions, making complex equations more approachable. It often involves substituting variables with known identities or simplifying expressions by rewriting them in alternative forms.
In this particular exercise, the transformation begins with substituting \(x = \tan \theta\) into our function \(f(x)\). With this substitution, the problem

• transform \(f(x)\) into \( \frac{2 \tan \theta}{1 + (\tan \theta)^2} \),

appearing somewhat complex initially. The Pythagorean identity is then used to further simplify, seamlessly bridging our way to apply the double-angle formula. The transformation process alters the problem into a guise that is manageable and, with skill, can be neatly resolved back to the original task of demonstrating that \( f(\tan \theta) \) equals \( \sin 2\theta \). Such transformations are invaluable in trigonometry and show the interconnectedness of its various identities.