$(\cot \alpha)^{\cos 2\alpha} \ge \frac{1}{\sin 2\alpha}$

Question

Let $ 0< \alpha < \dfrac{\pi}{2}$. Prove: $$(\cot \alpha)^{\cos 2\alpha} \ge \dfrac{1}{\sin 2\alpha}$$

Answer 1

do manipulations by taking logs and reduce it to x^x(1-x)^(1-x)>1/2.Again take log of tis.find derivative and find the local minima or maxima=1/1+e^2 and second derivative which is greater than zero hence minima. not put it in the function and estimate it