If $x,y,z\in (0,\frac{\pi}2 )$, find all solutions to: $$\begin{cases} \tan x+\sin y+\sin z=3x \\ \sin x+\tan y+\sin z=3y \\ \sin x+\sin y+\tan z=3z\end{cases}$$ This question was deleted here: https://math.stackexchange.com/questions/3747782/
My solution.
Let $f(x)=\tan{x}+2\sin{x}-3x$.
Thus, by AM-GM $$f'(x)=\frac{1}{\cos^2x}+2\cos{x}-3\geq3\sqrt[3]{\frac{1}{\cos^2x}\cdot(\cos{x})^2}-3=0,$$ which says that $f$ increases.
Thus, $$f(x)>f(0)=0$$ and $$3\sum_{cyc}x=\sum_{cyc}(\tan{x}+\sin{y}+\sin{z})=\sum_{cyc}(\tan{x}+2\sin{y})>3\sum_{cyc}x,$$ which is impossible.
Id est, our system has no solutions $(x,y,z)$, where $\{x,y,z\}\subset\left(0,\frac{\pi}{2}\right).$
Is there some alternative solution for this system?