Solution to inequality

Question

For $0<x<pi/2$ we have to find the minimum value of $$ \frac{sinx +cosx}{sinx+tanx} + \frac{sinx+cosx}{cosx+cotx} + \frac{tanx+cotx}{cosx+tanx} + \frac{tanx+cotx}{sinx+cotx}$$

Also a hint was given to make use of Engel's inequality. I tried by converting everything in terms of $sinx$ and $cosx$ but was not able to make any progress. How to tackle this question?

Answer 1

For $x=\frac{\pi}{4}$ we obtain a value $4$.

We'll prove that it's a minimal value.

Indeed, by AM-GM twice and by C-S we obtain: $$\frac{\sin x +\cos x}{\sin x+\tan x} + \frac{\sin x+\cos x}{\cos x+\cot x} + \frac{\tan x+\cot x}{\cos x+\tan x} + \frac{\tan x+\cot x}{\sin x+\cot x}\geq$$ $$\geq\frac{2(\sin{x}+\cos{x})}{\sqrt{(1+\sin{x})(1+\cos{x})}}+\frac{4(\tan{x}+\cot{x})}{\sin{x}+\cos{x}+\tan{x}+\cot{x}}\geq$$ $$\geq\frac{4(\sin{x}+\cos{x})}{2+\sin{x}+\cos{x}}+\frac{4}{\sin{x}\cos{x}(\sin{x}+\cos{x})+1}.$$ Now, let $\sin{x}+\cos{x}=t$.

Thus, by C-S again: $$0<t\leq\sqrt{(1+1)(\sin^2x+\cos^2x)}=\sqrt2.$$ Id est, it's enough to prove that: $$\frac{4t}{2+t}+\frac{4}{\frac{t^2-1}{2}\cdot t+1}\geq4$$ or $$t(t^2-2)\leq0$$ and we are done!