$\text{ If }\alpha +\beta ={90}^{\circ }\text{, then the minimum and maximum values of }\cos \alpha \cos \beta \text{ are }$

$\mathrm{Let} y=\cos \alpha \cos \beta \text{ then, }$

$y=\frac{1}{2}(2\cos \alpha \cos \beta )$

$=\frac{1}{2}[\cos (\alpha +\beta )+\cos (\alpha -\beta \left)\right]$

$=\frac{1}{2}\left[\cos {90}^{\circ }+\cos (\alpha -\beta )\right]$,  $\left[\because \alpha +\beta ={90}^{\circ }\right]$

$\Rightarrow y=\frac{1}{2}\cos (\alpha -\beta )$

$-1\le \cos (\alpha -\beta )\le 1$

$$\begin{gathered} \begin{array}{l}\Rightarrow \frac{-1}{2}\le y\le \frac{1}{2}\\ \Rightarrow \frac{-1}{2}\le \cos a\cos \beta \le \frac{1}{2}\end{array} \\ \therefore \text{ the maximum,minimum values of }\cos \alpha \cos \beta \text{ is }\frac{1}{2}\text{,-}\frac{1}{2} \end{gathered}$$