TITLE: Isolating $y$ in $\sin(xy)=\cos(xy)$
QUESTION [6 upvotes]: Given $\sin(xy)=\cos(xy)$, what is the best way to isolate $y$? Since $\sin(\frac{\pi}{2}) = \cos(\frac{\pi}{2})$ it would seem intuitive to say that $xy=\frac{\pi}{2}$ and thus that $y=\frac{\pi}{2x}$
Is this the correct approach, or am I missing something important?

REPLY [12 votes]: $$sin(xy)=cos(xy)\iff xy= \frac{\pi}{4}+k\pi;\space k\in \mathbb Z$$
Thus $$\color{red}{y=\frac{(4k+1)\pi}{4x}}$$