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

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Precalculus: the Mathematics of Numbers, Functions and Equations

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