Question

cot 4A = 1/cot 4A

Answer 1

$$cot(4A) = \frac{1}{cot(4A)}$$

First, let's multiply this equation by $\cot(4A)$, $\cot(4A)$ cannot be zero because it's in the denominator. Thus, we obtain:

$$cot^2 4A = 1, \quad \cot 4A \neq 0$$

Therefore

$$\cot 4A = \pm 1$$

From that, using well-known geometric formulae, we obtain that

$$4A = \frac{\pi}{4} + \frac{\pi k}{2}, \quad k \in \mathbb{Z}$$

or

$$A = \frac{\pi}{16} + \frac{\pi k}{8}, \quad k \in \mathbb{Z}$$