How prove this inequality $\sum\limits_{cyc}\frac{1-2\sin{\frac{C}{2}}}{\sin{\frac{B}{2}}}\ge 0$

Asked Mar 28 '14 at 12:58

Active Oct 20 '16 at 13:36

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in $\Delta ABC$,prove or disprove $$\dfrac{1-2\sin{\dfrac{C}{2}}}{\sin{\dfrac{B}{2}}}+\dfrac{1-2\sin{\dfrac{A}{2}}}{\sin{\dfrac{C}{2}}}+\dfrac{1-2\sin{\dfrac{A}{2}}}{\sin{\dfrac{A}{2}}}\ge 0$$

My idea: since $$\sum_{cyc}\dfrac{1-2\sin{\dfrac{C}{2}}}{\sin{\dfrac{B}{2}}}\ge 0$$ $$\sum_{cyc}\sin{\dfrac{A}{2}}\sin{\dfrac{C}{2}}(1-2\sin{\dfrac{C}{2}})\ge 0$$ then I can't

Thank you for you help

edited Oct 20 '16 at 13:36

Michael Rozenberg

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asked Mar 28 '14 at 12:58

math110

93,304

Should not the combination be cyclic like th last term will be $$\frac{1-2\sin\frac B2}{\sin\frac A2}$$ – lab bhattacharjee Mar 28 '14 at 14:17

How prove this inequality $\sum\limits_{cyc}\frac{1-2\sin{\frac{C}{2}}}{\sin{\frac{B}{2}}}\ge 0$

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