prova

$\displaylines{\sin(x)+\cos(x)\leq\frac{1}{\sin(x)}\iff \frac{1}{\sin(x)}-\sin(x)-\cos(x)\geq 0 \\ \iff \frac{1-\sin^2(x)-\sin(x)\cos(x)}{\sin(x)}\geq 0 \\ \iff \frac{\cos^2(x)-\sin(x)\cos(x)}{\sin(x)}\geq 0\iff\cot(x)(\cos(x)-\sin(x))\geq 0.}$