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

Question

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.

Answer 1

$\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}=\lim_{n\to\infty}\frac{\sum_{k=1}^{n}(cos k+sin k)}{\prod_{k=1}^{n}\cos k\sin k}=\lim_{n\to\infty}\frac{(cos 1+sin 2)+(cos 2+sin 2)+...+(cos n+sin n)}{(\cos 1\sin 1)(\cos 2\sin 2)...(\cos n\sin n)}\approx\lim_{n\to\infty}\frac{\cos n+\sin n}{\cos n\sin n}=\lim_{n\to\infty}(\frac{1}{\sin n}+\frac{1}{\cos n})$

I don't know how continue it