# How to prove $e^{\cos(1)} \sin(\sin(1)) = \sum_{n=1}^{\infty} \frac{\sin(n)}{n!}$?

Having the following equation:

$$e^{\cos(1)} \sin(\sin(1)) = \sum_{n=1}^{\infty} \frac{\sin(n)}{n!}$$

How can it be proven analytically? Left hand side and right hand side seems to be equal when testing numerically.

I have tried to use series but was unable to get any useful results.

The right-hand side is $$\Im\sum_{n\ge 0}\frac{e^{in}}{n!}=\Im\exp e^i=\Im\exp (\cos 1+i\sin 1)=\exp\cos 1\cdot\sin\sin 1.$$
• Very elegant!${}$ – Clayton Nov 7 '18 at 21:38