<s>[INST] Rearrangement inequality and minimal value of $\frac{\sin^3x}{\cos x} +\frac{\cos^3x}{\sin x}$ For $x \in \left(0, \dfrac{\pi}{2}\right)$, is the minimum value of $\dfrac{\sin^3x}{\cos x} +\dfrac{\cos^3x}{\sin x} = 1$? So considering ($\dfrac{1}{\cos x}$, $\dfrac{1}{\sin x}$) and ($\sin^3x$, $\cos^3x$), is it right to use the rearrangement inequality and conclude that $\dfrac{\sin^3x}{\cos x} +\dfrac{\cos^3x}{\sin x}$ is more than or equal to $\dfrac{\sin^3x}{\sin x} +\dfrac{\cos^3x}{\cos x}$ which is equal to $1$? Thanks. 
 [/INST] Cauchy Schwarz Inequality
$$\frac{\sin^3 x}{\cos x}+\frac{\cos^3 x}{\sin x}=\frac{\sin^4 x}{\cos x\sin x}+\frac{\cos^4 x}{\sin x\cos x}\geq \frac{(\sin^2 x+\cos^2 x)^2}{2\sin x\cos x}$$
$$\frac{\sin^3 x}{\cos x}+\frac{\cos^3 x}{\sin x}\geq \frac{1}{\sin 2x}\geq 1$$
equality hold when  $$\frac{\sin^2 x}{\sin x\cos x}=\frac{\cos^2 x}{\sin x\cos x}$$
that is $\displaystyle x=\frac{\pi}{4}.$
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