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A trigonometric identity is an equation that is true for all values of the variables it contains. When you encounter a trigonometric problem involving the difference of angles, such as the exercise asking for the value of \( \tan(u - v) \), you are dealing with one of these powerful identities.

The specific identity used in this case is the tangent difference formula, which expresses \( \tan(u - v) \) as a quotient of the difference of the tangents of \( u \) and \( v \) and the sum of their products. That is:\[ \tan(u - v) = \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)} \]

Understanding these identities is essential for simplifying and solving trigonometry problems, as they allow for the transformation of complex trigonometric expressions into simpler forms. Identities also help in verifying equations and proving various mathematical theorems.

The tan function, short for tangent, is one of the six fundamental trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, or as the sine function divided by the cosine function:

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

The importance of the tan function in trigonometry cannot be overstated. It's particularly useful in situations involving angles of inclination, slopes, and whenever perpendicular and base aspects of a scenario are involved. The tan function is unique in that it's periodic with a period of \( \pi \), and has asymptotes, points where the function tends towards infinity, which occur at \( \frac{\pi}{2} \) and multiples thereof.

Problem-solving in trigonometry often involves first identifying the relevant formulas and identities, as well as understanding the relationships between the trigonometric functions. It typically includes several steps, starting with outlining what is given and what needs to be found.

For example, in our given exercise, you are tasked with finding the expression for \( \tan(u - v) \). The first step is recognizing the appropriate identity to use, applying it, and then simplifying if necessary. It’s essential to also understand the domain and range of the trigonometric functions involved, as they have certain restrictions depending on the quadrant the angles are in.

Mastering trigonometry problem solving opens up pathways to advanced mathematics, including calculus and further analytical geometry, and also has applications in physics, engineering, and even computer science.