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Understanding how to express the cotangent function in terms of sine and cosine is a fundamental aspect of trigonometry, which helps in simplifying many trigonometric expressions and verifying identities. The cotangent of an angle in a right-angled triangle is the ratio of the adjacent side to the opposite side. In trigonometry, this can be represented as the ratio of cosine to sine.

Specifically, if you have an angle denoted as \( t \), then the cotangent of \( t \) is described by the formula:\[ \cot(t) = \frac{\cos(t)}{\sin(t)} \]. This formula can be used to rewrite trigonometric expressions involving cotangent in terms of sine and cosine, which can be particularly helpful when you need to solve equations or verify identities.

One of the most powerful tools at our disposal in trigonometry is the Pythagorean identity, an equation that relates the squares of sine and cosine functions. Derived from the Pythagorean theorem, the identity is a cornerstone for solving trigonometric problems and verifying identities.

The basic form of the Pythagorean identity is:\[ \sin^2(t) + \cos^2(t) = 1 \].
Here, for any angle \( t \), the square of the sine of \( t \), plus the square of the cosine of \( t \), always equals one. When we need to simplify trigonometric expressions or find alternate forms of an expression, this identity is an invaluable shortcut. By rearranging the identity, we can express \( \cos^2(t) \) as \( 1 - \sin^2(t) \) or \( \sin^2(t) \) as \( 1 - \cos^2(t) \), as was done in the given exercise to show that \( \cos(t) \cot(t) \) is equal to \( \frac{1-\sin^2(t)}{\sin(t)} \).

Verifying trigonometric identities can at times be a challenging task, but it's a critical skill for understanding trigonometry's foundational principles. An identity in trigonometry is an equation that holds true for all values of the involved variable where both sides of the equation are defined.

To verify an identity, one approach is to transform one or both sides of the equation until they appear structurally identical. Start by working with one side of the identity, then use various algebraic and trigonometric operations—including the fundamental identities like the Pythagorean identity—to simplify it. Sometimes, expressing functions in different forms, as in the given exercise where cotangent is expressed in terms of sine and cosine, can lead to the solution.

In the given problem, after simplifying the left side using the cotangent's expression and the Pythagorean identity, it became identical to the right side. Thus, we verify that the given equation is indeed an identity. This process of simplification and substitution offers a systematic way to verify a wide range of trigonometric identities.