$\cos (\alpha -\beta )=1\text{ and }\cos (\alpha +\beta )=1/e,$where $\alpha ,\beta \in [-\pi ,\pi ]$ Number of pairs of $\alpha ,\beta$ which satisfy both the equations is ____ 

$\text{ Now }\cos (\alpha -\beta )=1\Rightarrow \alpha -\beta =0\text{ or }\alpha =\beta$

$\Rightarrow \therefore \cos (\alpha +\beta )=1/e\Rightarrow \cos 2\alpha =1/e$

$\Rightarrow \therefore 0<1/e<1\text{ and }2\alpha \in [-2\pi ,2\pi ]$

$\text{ There will be two values of }2\alpha \text{ satisfying }$

$\cos 2\alpha =1/e\text{ in }[0,2\pi ]\text{ and two in }[-2\pi ,0]$

$\text{ No, of pairs of }(\alpha ,\beta )=4$