<s>[INST] Trigonometric inequality in an obtuse triangle Let $ABC$ be an obtuse triangle with $A$ the obtuse angle. I conjecture that the following inequality is true $$\sin B + \sin C \le |\tan A|.$$ 
Show that it holds or give a counterexample.
 [/INST] Since $B ,C , B+C <   \frac{\pi}{2} $ we have $\frac{\cos B }{\cos (B +C )} >1$ and $\frac{\cos C }{\cos (B+C )} >1$ hence $$|\tan A|=\tan (B+C) =\frac{\cos B}{\cos (B+C)} \cdot \sin C + \frac{\cos C}{\cos (B+C)} \cdot \sin B >1\cdot \sin C +1\cdot \sin B =\sin C +\sin B .$$ 
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