If a and b are positive quantities such that a > b , then minimum value of  $\mathrm{asec}\mathrm{\theta }-\mathrm{btan}\mathrm{\theta }$ is

$\text{ Let }\mathrm{s}=\mathrm{asec}\mathrm{\theta }-\mathrm{btan}\mathrm{\theta }$

$\begin{array}{l}\mathrm{or} \mathrm{btan}\mathrm{\theta }+\mathrm{s}=\mathrm{asec}\mathrm{\theta }\\ \mathrm{or} \left({\mathrm{a}}^2-{\mathrm{b}}^2\right){\tan }^2\mathrm{\theta }-2\mathrm{bstan}\mathrm{\theta }+\left({\mathrm{a}}^2-{\mathrm{s}}^2\right)=0\end{array}$

$\text{ For }\tan \mathrm{\theta }\text{ to be real }4{\mathrm{b}}^2{\mathrm{s}}^2-4\left({\mathrm{a}}^2-{\mathrm{b}}^2\right)\left({\mathrm{a}}^2-{\mathrm{s}}^2\right)\ge 0$

$\begin{array}{l}\mathrm{or} {\mathrm{a}}^2{\mathrm{s}}^2\ge {\mathrm{a}}^2\left({\mathrm{a}}^2-{\mathrm{b}}^2\right)\\ \mathrm{or} \mathrm{s}\ge \sqrt{{\mathrm{a}}^2-{\mathrm{b}}^2}\end{array}$

Therefore, the minimum value of s is $\sqrt{{\mathrm{a}}^2-{\mathrm{b}}^2}$