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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The word itself comes from the Greek 'trig艒non' (triangle) and 'metron' (measure). It has applications in various fields such as physics, engineering, astronomy, and even in the construction of buildings and creation of maps.

The primary functions in trigonometry are sine, cosine, and tangent, which are ratios that relate the angles of a right triangle to its sides. Understanding trigonometry is crucial for solving problems involving right-angled triangles and for dealing with periodic phenomena鈥攁ny occurrences that repeat at regular intervals, like sound waves and tides.

The cosine function, often abbreviated as 'cos', is one of the fundamental trigonometric functions. It relates the angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. In mathematical terms, if \theta is an angle in a right triangle, and the lengths of the adjacent side and hypotenuse are a and c, respectively, then the cosine of \theta is defined as \( \cos(\theta) = \frac{a}{c} \).

Cosine can also be understood as a wave function in the context of periodic phenomena, with a value ranging from -1 to 1. It is an even function, which means that its graph is symmetrical about the y-axis, and it has a period of 2\(\pi\) radians, repeating its values every 2\(\pi\) radians or 360 degrees.

Trigonometric identities are equations that hold true for all values of the variables involved. These identities are used to simplify trigonometric expressions, solve equations, and prove other mathematical statements. They are essentially the backbone of trigonometric problem-solving.

The Cosine Difference Identity, \( \cos(u-v) = \cos(u)\cos(v) + \sin(u)\sin(v) \), for instance, is extremely useful when you need to determine the cosine of the difference between two angles. This identity demonstrates that the cosine of a difference can actually be broken down into a product of sines and cosines of individual angles, which often simplifies calculations in problems involving trigonometry. Knowing such identities and how to derive them enriches the problem-solving toolkit of students in mathematics and related fields.