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 $\cos ^{-1} \mathrm{x}$ takes values in $[0, \pi]$, which of the following options is/are correct?
Select ALL correct options:
A
$\sin \left(7 \cos ^{-1} f(5)\right)=0$
B
$f(4)=\frac{\sqrt{3}}{2}$
C
$\lim _{n \rightarrow \infty} f(n)=\frac{1}{2}$
D
If $\alpha=\tan \left(\cos ^{-1} f(6)\right)$, then $\alpha^2+2 \alpha-1=0$
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