<s>[INST] Find the least value of $4\csc^{2} x+9\sin^{2} x$ 
Find the least value of $4\csc^{2} x+9\sin^{2} x$

$a.)\ 14 \ \ \ \ \ \ \ \ \ b.)\ 10 \\
c.)\ 11 \ \ \ \ \ \ \ \ \ \color{green}{d.)\ 12} $
$4\csc^{2} x+9\sin^{2} x \\
= \dfrac{4}{\sin^{2} x} +9\sin^{2} x \\
= \dfrac{4+9\sin^{4} x}{\sin^{2} x}  \\
= 13  \ \ \ \ \ \ \ \ \ \ \ \ (0\leq \sin^{2} x\leq 1)
$
But that is not in options.
I look for a short and simple way.
I have studied maths upto $12$th grade.
 [/INST] $$\left(\frac{2}{\sin x}-3\sin x\right)^2\geq0$$
$$\iff\csc^2 x+9\sin^2 x\geq12$$ with equality achieved when $\sin^2x=2/3.$
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