If $\sec \theta + \tan \theta = p,$ then $\tan \theta$ is equal to

The value of $\tan 81^{\circ} - \tan 63^{\circ} - \tan 27^{\circ} + \tan 9^{\circ}$ is

The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin (x + y)$ and $|x| + |y| = 1$ is

The expression $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$ is equal to

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 $\alpha^2+2 \alpha-1=0$

If $f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x}$ for all $x \in R$,then $f(2023) = $