Finding minimum value of $a^2\cot (9^\circ)+b^2\cot (27^\circ)+c^2\cot(63^\circ)+d^2\cot(81^\circ)$ subjected to the condition $a+b+c+d=5,a,b,c,d$ all are real number
Try: Cauchy Schwarz Inequality
$$(a^2+b^2+c^2+d^2)\bigg(\cot^2(9^\circ)+\cot^2(27^\circ)+\cot^2(63^\circ)+\cot^2(81^\circ)\bigg)\geq \bigg(a\cot (9^\circ)+b\cot (27^\circ)+c\cot(63^\circ)+d\cot(81^\circ)\bigg)^2$$
Could some help me how to solve, thanks
Best Answer
Hint
$$(a^2\cot \,9°+b^2\cot \,27°+c^2\cot \,63°+d^2\cot \,81°)(\tan \,9^\circ+\tan \,27^\circ+\tan \,63^\circ+\tan \,81^\circ)\ge \left (\sum a\right)^2= 25 $$