The least value of $$(4sec^2\theta + 9cosec^2\theta)$$ is

$$4sec^2\theta+9cosec^2\theta$$
or $$4+4tan^2\theta+9+9cot^2\theta$$
or $$13+4tan^2\theta+9cot^2\theta$$
or $$ 13+4tan^2\theta+\frac{9}{tan^2\theta} $$
or $$  13-12+(2tan\theta+\frac{3}{tan\theta})^2 $$    (eq. (1) )
or now above expression to be minimum, equation $$(2tan\theta+\frac{3}{tan\theta})^2$$ should be minimum.
So applying $$A.M.\geq G.M. $$
$$\frac{(2tan\theta +\frac{3}{tan\theta})}{2} \geq \sqrt{6}$$
or $${(2tan\theta+\frac{3}{tan\theta})}=2\sqrt{6}$$ ( for value to be minimum)
After putting above value in eq.(1) , we will get least value of expression as 25.