Question

Find the general solution of the following equation:

\[\tan mx + \cot nx = 0\]

Answer 1

We have:

\[\tan mx + \cot nx = 0\]

\[\Rightarrow \tan mx = - \cot nx\]

\[ \Rightarrow \tan mx = \tan \left( \frac{\pi}{2} + nx \right)\]

\[ \Rightarrow mx = r\pi + \left( \frac{\pi}{2} + nx \right), r \in Z\]

\[ \Rightarrow (m - n) x = r\pi + \frac{\pi}{2}, r \in Z\]

\[ \Rightarrow x = \left( \frac{2r + 1}{m - n} \right)\frac{\pi}{2}, r \in Z\]