Computing the limit of $n cdot arccos left( left(frac{n^2-1}{n^2+1}right)^{cos (1/n)} right)$

I need to solve this earth's wonder:

$$\lim_{n \rightarrow \infty} \left[n \; \arccos
\left( \left(\frac{n^2-1}{n^2+1}\right)^{\cos \frac{1}{n}} \right)\right]$$

I have tried to write down it using $e^{v \ln u}$,and then used L'Hôpital's rule, but with no luck, i'm constantly getting indeterminate form like $\infty-\infty$ inside $\ln$ which makes another application of L'Hôpital impossible.

My professor told me (with great smile on its face) that if i use Taylor expansion, it will lead me into the abyss...

any hints about possible rewriting this limit and what i should use would be VERY helpful. Thanks in advance.