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Trigonometric identities are like the DNA of trigonometry—a set of fundamental formulas that define relationships between different trigonometric functions. These equations are used to simplify complex expressions and solve trigonometric equations. They include the Pythagorean identities, sum and difference formulas, double angle formulas, and more.
For example, the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) expresses a relationship between the sine and cosine of an angle. Understanding these identities is key to mastering trigonometry because they allow us to manipulate and transform trigonometric expressions into more workable forms. They also play a critical role in solving exercises like the one in this article, where the tangent of the difference between two angles needs to be found.

The tangent function, often denoted as \(\tan\text{ of an angle}\), is a significant trigonometric function that represents the ratio of the opposite side to the adjacent side of a right triangle. It's one of the three primary trigonometric functions, along with sine and cosine.
In practical problems, the tangent function can help determine steepness or incline and is particularly useful in fields such as physics, engineering, and architecture. A deep understanding of the tangent function, its graph, and properties, such as its periodicity and symmetry, is crucial for solving trigonometric problems efficiently.

The angle subtraction formula for tangent is a powerful tool in trigonometry that enables us to express the tangent of the difference between two angles in terms of the tangents of the individual angles. The formula is \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\), where \(a\) and \(b\) represent angles.
This formula is extremely helpful when the exact values of the tangents are either known or can be easily computed. It simplifies what might otherwise be a very complex trigonometric problem. In our exercise, the use of the angle subtraction formula transforms the problem into a simpler format, resulting in the calculation of \(\tan(70^\circ) - \tan(20^\circ)\) which further boils down, with the step-by-step approach, to an undefined value due to a zero in the denominator, signifying an infinite tangent, or in other words, a vertical line.