Simplify $|y \cdot \cos(xy)|\cdot \frac{|x|}{|\sin(xy)|}$

Question

$$\begin{eqnarray}&&|y \cdot \cos(xy)|\cdot \frac{|x|}{|\sin(xy)|} \\ &= &\frac{|x| \cdot |y \cdot \cos(xy)|}{|\sin(xy)|}\\ &= &\frac{|xy \cdot \cos(xy)|}{|\sin(xy)|}\\ &= &\left|\frac{xy \cdot \cos(xy)}{\sin(xy)}\right|\\ &= &\left|\frac{xy}{\tan(xy)}\right|\end{eqnarray}$$

I need to simplfy it somehow because it's used in a very big task which might get more complicated if I keep it unsimplified. I hope it's fine like that?

Answer 1

It is ''fine'' but with some attention. The starting function is not defined for $xy=k\pi$, the final function is not defined also for $xy=\frac{\pi}{2}+k\pi$, so this case require a special consideration.

Answer 2

If you're worried about a calculation taking too long and if $|xy|$ is small, you can use the Laurant series for $\cot t$ centered at $t=0$. Your last expression is

$$= |xy||\cot xy | = |xy|\left| \frac{1}{xy}-\frac{xy}{3}-\frac{(xy)^3}{45} - \frac{2(xy)^5}{945} - \cdots \right| = \left| 1-\frac{(xy)^2}{3} - \frac{(xy)^4}{45} - \cdots \right|. $$