Question Number 173820 by dragan91 last updated on 18/Jul/22 $$ \\ $$ Prove $$ \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{7}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{8}\pi}{\mathrm{19}}}+\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\mathrm{2}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{3}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{5}\pi}{\mathrm{19}}}− \\ $$$$−\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\mathrm{4}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{6}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{9}\pi}{\mathrm{19}}}=\frac{\mathrm{1}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{19}}−\mathrm{1}} \\ $$$$ \\ $$$$ \\ $$$$ \\…
Question Number 108270 by Muhsang S L last updated on 15/Aug/20 $$\mathrm{If}\:{x},\:{y},\:{z}\:>\:−\mathrm{1},\:\mathrm{show}\:\mathrm{that}\:\frac{\mathrm{1}\:+\:{x}^{\mathrm{2}} }{\mathrm{1}\:+\:{y}\:+\:{z}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}\:+\:{y}^{\mathrm{2}} }{\mathrm{1}\:+\:{z}\:+\:{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}\:+\:{z}^{\mathrm{2}} }{\mathrm{1}\:+\:{x}\:+\:{y}^{\mathrm{2}} }\:\geqslant\:\mathrm{2} \\ $$ Commented by udaythool last updated…
Question Number 173815 by mr W last updated on 18/Jul/22 $${prove}\: \\ $$$$\left(\frac{{n}+\mathrm{1}}{\mathrm{3}}\right)^{{n}} <{n}! \\ $$ Commented by mr W last updated on 19/Jul/22 $${good}\:{idea}\:{sir}!\:{thanks}!…
Question Number 173798 by payamnazerian last updated on 18/Jul/22 $$ \\ $$$$\:\:\:\:\:\:\:\left(\:\varphi^{\:\mathrm{5}} +\:\varphi^{\:\mathrm{4}} +\:\mathrm{1}\right)^{\:\mathrm{2}} =\:{m}\varphi\:+\:{n} \\ $$$$\:\:\:\:\:\:\:\:\:{m},{n}=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…….. \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$ Answered by…
Question Number 42719 by jbm last updated on 01/Sep/18 Commented by Joel578 last updated on 02/Sep/18 $$\mathrm{Sophie}\:\mathrm{Germain} \\ $$$${p}^{\mathrm{4}} \:+\:\mathrm{4}{q}^{\mathrm{4}} \:=\:\left({p}^{\mathrm{2}} \:+\:\mathrm{2}{pq}\:+\:\mathrm{2}{q}^{\mathrm{2}} \right)\left({p}^{\mathrm{2}} \:−\:\mathrm{2}{pq}\:+\:\mathrm{2}{q}^{\mathrm{2}} \right)…
Question Number 173787 by mnjuly1970 last updated on 18/Jul/22 $$\:\mathrm{4}{sin}^{\:\mathrm{2}} \left({x}\right)−\:\mathrm{4}{sin}\left({x}\right)={cos}^{\:\mathrm{2}} \left({x}\right)−\mathrm{2}{cos}\left({x}\right) \\ $$$$\:\:\:,\:\mathrm{0}<{x}<\frac{\pi}{\mathrm{2}}\:.\:\:\:{Find}\:\:\:\:\:\:\:{sin}\left({x}\right)=? \\ $$ Commented by infinityaction last updated on 18/Jul/22 $$\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}} \\…
Question Number 173776 by AgniMath last updated on 18/Jul/22 Answered by thfchristopher last updated on 18/Jul/22 $$\mathrm{When}\:{n}=\mathrm{1}, \\ $$$$\mathrm{R}.\mathrm{H}.\mathrm{S}.=\frac{\mathrm{1}×\mathrm{2}}{\mathrm{2}} \\ $$$$=\mathrm{1} \\ $$$$=\mathrm{L}.\mathrm{H}.\mathrm{S}. \\ $$$$\mathrm{Assume}\:{n}={k}\:\mathrm{is}\:\mathrm{true},…
Question Number 108238 by Study last updated on 15/Aug/20 $${y}={e}^{{x}} {ln}\left({sin}\mathrm{2}{x}\right)\:\:\:\:\:\:\:\:\:\:\:\frac{{dy}}{{dx}}=?? \\ $$ Answered by Dwaipayan Shikari last updated on 15/Aug/20 $$\frac{{dy}}{{dx}}={e}^{{x}} {log}\left({sin}\mathrm{2}{x}\right)+\mathrm{2}{e}^{{x}} \frac{{cos}\mathrm{2}{x}}{{sin}\mathrm{2}{x}} \\…
Question Number 108237 by Study last updated on 15/Aug/20 $${y}=\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}−{ln}\left(\frac{\mathrm{1}}{{x}}+\sqrt{\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}\right) \\ $$$$\frac{{dy}}{{dx}}=? \\ $$ Answered by Dwaipayan Shikari last updated on 15/Aug/20 $${y}=\sqrt{{x}^{\mathrm{2}}…
Question Number 108219 by mr W last updated on 15/Aug/20 Commented by mr W last updated on 15/Aug/20 $${an}\:{unsolved}\:{old}\:{question}\:{reposted} \\ $$ Commented by john santu…