Find the exact value of the expression. \( \cot \left(\frac{\pi}{4}\right)=\square \)

The value of \( \cot \left(\frac{\pi}{4}\right) \) is 1. This is because the cotangent function is defined as the ratio of the cosine to the sine of an angle, or \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). At \( \frac{\pi}{4} \), both sine and cosine are equal, specifically \( \sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). Thus, \( \cot\left(\frac{\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \). Interestingly, this value is also crucial in various fields, such as trigonometry, physics, and engineering, where it often appears in problems involving right triangles or oscillatory motion. Understanding the unit circle and how angles correspond to specific coordinates helps reinforce why \( \cot\left(\frac{\pi}{4}\right) = 1 \) is not only a lovely mathematical fact but a building block for more advanced concepts!