91Ó°ÊÓ

The cotangent function, often written as \(\text{cot} \theta\), is one of the fundamental trigonometric functions. It is closely related to the tangent function. In fact, \(\text{cot} \theta\) is simply the reciprocal of \(\text{tan} \theta\). That means \(\text{cot} \theta = \frac{1}{\text{tan} \theta}\).

The tangent function, represented as \(\text{tan} \theta\), is another key trigonometric function. It expresses the ratio of the sine and cosine of an angle: \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\). The tangent function has a periodic nature and plays a crucial role in various trigonometric identities and equations.

The double angle formula is a powerful tool in trigonometry, allowing us to express functions of double angles in terms of single angles. For the tangent function, the double angle formula is given by: \[ \text{tan} \(2x\) = \frac{2 \text{tan} \(x\)}{1 - \text{tan}^{2} \(x\)} \] By using this formula, we can simplify and solve more complex trigonometric expressions, such as when verifying trigonometric identities. For example, to find \(\text{tan} \(4 \theta\)\), we can first find \(\text{tan} \(2 \theta\)\) and then apply the formula recursively: \[ \text{tan} \(4 \theta\) = \frac{2 \text{tan} \(2 \theta\)}{1 - \text{tan}^{2} \(2 \theta\)} \] This approach simplifies the process of verifying identities like the one given in the problem statement.