sequences and series - Find $lim_{ntoinfty}frac{sum_{k=1}^{n}cos k+sum_{k=1}^{n}sin k}{prod_{k=1}^{n}cos ksin k}$

Find the following limit:

$$\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$$

The numerator can be simplified by using Euler's formula and the sum of geometric series. I am struggling on the denomenator. How can we simplify that product? By the way, I don't even know whether or not this limit exist.