For non-negative integers n,let
$$f(n) = \frac{{\sum_{k = 0}^n}\sin \left(\frac{k + 1}{n + 2}\pi\right) \sin \left(\frac{k + 2}{n + 2}\pi\right)}{\sum_{k = 0}^n \sin^2 \left(\frac{k + 1}{n + 2}\pi\right)} $$
Assuming $$\cos^{-1}x$$ takes values in $$[0,\pi],$$ which of the following options is/are correct?

A

$$f(4) = \frac{\sqrt 3}{2}$$

B

$$\lim_{n \rightarrow \infty} f(n) = \frac{1}{2}$$

C

If $$\alpha = \tan (\cos^{-1} f (6)),$$ then $$\alpha^2 + 2 \alpha - 1 = 0$$

D

$$\sin(7 \cos^{-1} f(5)) = 0$$