If $3 \cos \theta + 4 \sin \theta = 5$,then the value of $3 \sin \theta - 4 \cos \theta$ is:

If $f(\theta) = \cos^3 \theta + \cos^3 \left(\frac{2\pi}{3} + \theta\right) + \cos^3 \left(\theta - \frac{2\pi}{3}\right)$,then $f\left(\frac{\pi}{5}\right) = $

The value of $\cos^2 \frac{\pi}{12} + \cos^2 \frac{\pi}{4} + \cos^2 \frac{5\pi}{12}$ is

$\operatorname{coth}^2 x - \tanh^2 x =$

If $5 \sinh x - \cosh x = 5$,then one of the values of $\tanh x$ is

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?
$(1)$ $\sin \left(7 \cos ^{-1} f(5)\right)=0$
$(2)$ $f(4)=\frac{\sqrt{3}}{2}$
$(3)$ $\lim _{n \rightarrow \infty} f(n)=\frac{1}{2}$
$(4)$ If $\alpha=\tan \left(\cos ^{-1} f(6)\right)$,then $\alpha^2+2 \alpha-1=0$