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

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

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?

• Yes, your simplification looks fine. But then again we do not know if will be useful in your computation, since we do not know what you are computing. Jun 12, 2017 at 20:10

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.
• Hmm do you think it's better to keep it like that, unsimplified? I shall be as precise as possible. Edit: $x,y \neq 0$ Jun 12, 2017 at 20:16
• No, the simplification works well, but you have to consider separately the case $xy= \frac{\pi}{2}+k \pi$. In this case the value of the function is $0$ because $\cos xy=0$ and consider that the condition $x,y \ne 0$ is not sufficient to avoid problems with the final function. Jun 12, 2017 at 20:21
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
$$= |xy||\cot xy | = |xy|\left| \frac{1}{xy}-\frac{xy}{3}-\frac{(xy)^3}{45} - \frac{2(xy)^5}{945} - \cdots \right| = \left| 1-\frac{(xy)^2}{3} - \frac{(xy)^4}{45} - \cdots \right|.$$