Question

Find the exact value of $\cot(\frac{\pi}{7}) + \cot(\frac{2\pi}{7}) - \cot(\frac{3\pi}{7})$.

Answer 1

$\sqrt{7}$

Answer 2

Let the given expression be $E$. We have $E = \cot\left(\frac{\pi}{7}\right) + \cot\left(\frac{2\pi}{7}\right) - \cot\left(\frac{3\pi}{7}\right)$ We know that for $\alpha = \frac{\pi}{7}$, we have $7\alpha = \pi$. This implies: $\cot(4\alpha) = \cot(\pi - 3\alpha) = -\cot(3\alpha)$ $\cot(5\alpha) = \cot(\pi - 2\alpha) = -\cot(2\alpha)$ $\cot(6\alpha) = \cot(\pi - \alpha) = -\cot(\alpha)$

The expression can be rewritten by substituting $-\cot(3\alpha) = \cot(4\alpha)$: $E = \cot\left(\frac{\pi}{7}\right) + \cot\left(\frac{2\pi}{7}\right) + \cot\left(\frac{4\pi}{7}\right)$ This is a specific case of a known identity related to the cotangents of angles in an arithmetic progression. For $n=7$, the sum $\cot(\frac{\pi}{n}) + \cot(\frac{2\pi}{n}) + \cot(\frac{4\pi}{n})$ is equal to $\sqrt{n}$.

Therefore, $E = \cot\left(\frac{\pi}{7}\right) + \cot\left(\frac{2\pi}{7}\right) + \cot\left(\frac{4\pi}{7}\right) = \sqrt{7}$ The exact value of the expression is $\sqrt{7}$.