The minimum value of ${\mathrm{atan}}^2\mathrm{x}+{\mathrm{bcot}}^2\mathrm{x}$  equals the maximum value of ${\mathrm{asin}}^2\mathrm{\theta }+{\mathrm{bcos}}^2\mathrm{\theta }\text{ where }\mathrm{a}>\mathrm{b}>0$ then $\frac{\mathrm{a}}{\mathrm{b}}$ is

Minimum value of ${\mathrm{atan}}^2\mathrm{x}+{\mathrm{bcot}}^2\mathrm{x}\text{ is }2\sqrt{\mathrm{ab}}$

And maximum value of ${\mathrm{asin}}^2\mathrm{\theta }+{\mathrm{bcos}}^2\mathrm{\theta }\text{ is }\mathrm{a}\text{. }$

$\left(\because {\mathrm{asin}}^2\mathrm{\theta }+{\mathrm{bcos}}^2\mathrm{\theta }=(\mathrm{a}-\mathrm{b}){\sin }^2\mathrm{\theta }+\mathrm{b}\right)$

$\text{ According to question, }a=2\sqrt{ab}$

$\therefore \mathrm{a}=4\mathrm{b}$