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

Now $\cos (\mathrm{\alpha }-\mathrm{\beta })=1\Rightarrow \mathrm{\alpha }-\mathrm{\beta }=0$
$\begin{array}{l}\Rightarrow \therefore \cos (\mathrm{\alpha }+\mathrm{\beta })=1/\mathrm{e}\Rightarrow \cos 2\mathrm{\alpha }=1/\mathrm{e}\\ \Rightarrow \therefore 0<1/\mathrm{e}<1\text{ and }2\mathrm{\alpha }\in [-2\mathrm{\pi },2\mathrm{\pi }]\end{array}$
There will be two values of 2α satisfying
$\cos 2\mathrm{\alpha }=1/\mathrm{e}$   in  $[0,2\mathrm{\pi }]\text{ and two in }[-2\mathrm{\pi },0]$
No, of pairs of (α,ß) = 4