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Sine and cosine are fundamental trigonometric functions that describe relationships in right-angled triangles. They are essential in expressing other trigonometric identities and simplifying expressions.
The sine function, denoted as \( \sin \theta \), gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. Similarly, the cosine function, \( \cos \theta \), measures the ratio of the adjacent side to the hypotenuse.

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

These two functions form the basis from which we derive other trigonometric identities. In the provided solution, converting tangent and secant to sine and cosine simplifies the given identity. This conversion makes it easier to follow through and verify the mathematical identity by expressing all terms in terms of the fundamental functions.

The tangent and secant functions are derived from sine and cosine, providing additional tools for analyzing angles and sides in a triangle.
The tangent function, \( \tan \theta \), is the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This expresses how the vertical component (opposite) relates to the horizontal component (adjacent) of a triangle. In identity verifications, converting from tangent to sine and cosine is common practice.

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Likewise, the secant function, \( \sec \theta \), is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \). It often finds its place in identities, especially where cosine appears in the denominator.

• \( \sec \theta = \frac{1}{\cos \theta} \)

By expressing both tangent and secant in terms of sine and cosine, the original trigonometric identity \(\frac{\tan ^{2} \theta}{\sec \theta} = \sin \theta \tan \theta\) becomes much simpler to handle, making it an easier task to equate both sides during verification.

Mathematical proofs, especially in trigonometry, involve showing that two expressions are equivalent using logical reasoning and identities. Verifying a trigonometric identity requires demonstrating that two expressions represent the same quantity for all values of the involved angle.
In this exercise, the problem starts with transforming terms into their sine and cosine equivalents. This approach is often a decisive first step. By transforming and simplifying, we can arrive at an equivalent expression on both sides of the equation.
The original identity to prove is \(\frac{\tan ^{2} \theta}{\sec \theta} = \sin \theta \tan \theta\). By substituting, simplifying, and converting back, we confirm both sides of the equation match, completing the proof.

• Identify trigonometric identities and substitutions
• Simplify step by step
• Confirm both expressions are equivalent

Proofs help establish certainty in mathematics, ensuring that identities hold true universally and are not just coincidental findings. By breaking down and understanding each step, the process demonstrates the logical structure behind the identity.