For the hyperbola $\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{{\sin }^2\alpha }=1$  which of the following remains constant when $\alpha$  varies.

$\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{{\sin }^2\alpha }=1$     we have ${\sin }^2\alpha ={\cos }^2\alpha \left(e^2-1\right)$ 

${\tan }^2\alpha =e^2-1$

$\therefore e=\sec \alpha$ 

Equation of directrix is $x=\frac{a}{e}$

$\Rightarrow x=\frac{\cos \alpha }{sec\alpha }\Rightarrow x={\cos }^2\alpha ---\left(2\right)$

Vertices $=\left(\pm \cos \alpha ,0\right)$        Foci $=(\pm \cos \alpha \text{\hspace{0.17em}.}\sec \alpha , 0)=(\pm 1,0)$