Minimum value Of trigonometry expression FIND THE MIN VALUE OF 4 cosec^2 x + 9 sin^2 x ? Please explain by both calculus and non-calculus methods ?

Another way using AM-GM inequality: for positive $a,b$ 
$$ \frac{a+b}{2} \geq \sqrt{ab} $$
Notice
$$ 4 \csc^2 x + 9 \sin^2 x = (2 \csc x)^2 + (3 \sin x)^2 \geq 2 \cdot 2 \csc x \cdot 3 \sin x = 12 \frac{1}{ \sin x}{\sin x} = 12$$
Therefore, we conclude that the minimum value is $12$
