<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] You already have an answer for (1). For (2), note by Cauchy Schwarz inequality,
$$LHS = \sum_{cyc} \frac{c^2}{c^2(a+b)}+\frac{(\sqrt[3]{abc})^2}{2abc} \ge \frac{(a+b+c+\sqrt[3]{abc})^2}{\sum_{cyc} c^2(a+b) + 2abc}$$
and further the denominator factorises as
$$\sum_{cyc} c^2(a+b) + 2abc = (a+b)(b+c)(c+a)$$
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