<s>[INST] Cauchy-Schwarz inequality problem The problems:



*Prove that $$\frac{\sin^3 a}{\sin b} + \frac{\cos^3 a}{\cos b} \geqslant \sec (a-b),$$ for all $a,b \in \bigl(0,\frac{\pi}{2}\bigr)$.


*Prove that $$\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} + \frac{1}{2\sqrt[3]{abc}} \geqslant \frac{(a+b+c+\sqrt[3]{abc})^2}{(a+b)(b+c)(c+a)},$$ for all $a,b,c > 0$.

Can someone give me a hint for the two problems. They are all based on the Cauchy-Schwarz inequality.
Just a hint.
 [/INST] One form of the Cauchy-Schwarz Inequality is
$$\sum_i^n \frac{x_i^2}{y_i}\ge \frac{\left(\sum_i^n x_i\right)^2}{\sum_i^n y_i}\tag 1$$
Let $n=2$ and $x_1=\sin^2a$, $x_2=\cos^2a$, $y_1=\sin a\sin b$, and $y_2=\cos a\cos b$
The result follows immediately after expanding $\sec (a-b)=\frac{1}{\cos a\cos b+\sin a\sin b}$  
To see this explicitly, we use $(1)$ and write
$$\begin{align}
\sum_i^n \frac{x_i^2}{y_i}&=\frac{\sin^4 a}{\sin a \sin b}+\frac{\cos^4 a}{\cos a\cos b}\\\\
&=\bbox[5px,border:2px solid #C0A000]{\frac{\sin^3a}{\sin b}+\frac{\cos^3 a}{\cos b}}\\\\
&\ge \frac{\left(\sum_i^n x_i\right)^2}{\sum_i^n y_i}\\\\
&=\frac{\sin^2 a +\cos^2 a}{\sin a\sin b+\cos a\cos b}\\\\
&=\frac{1}{\cos(a-b)}\\\\
&=\bbox[5px,border:2px solid #C0A000]{\sec(a-b)}
\end{align}$$
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