Question Number 195157 by Erico last updated on 25/Jul/23 $$\mathrm{Prove}\:\mathrm{that} \\ $$$$\frac{{x}^{\mathrm{3}} }{\mathrm{2}{sin}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}{arctan}\:\frac{{x}}{{y}}\right)}+\frac{{y}^{\mathrm{3}} }{\mathrm{2}{cos}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}{arctan}\:\frac{{y}}{{x}}\right)}=\left({x}+{y}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$ Answered by Frix last updated…
Question Number 195124 by Shlock last updated on 25/Jul/23 Answered by witcher3 last updated on 25/Jul/23 $$\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}\leqslant\sqrt{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{y}+\mathrm{1}\right)} \\ $$$$\Leftrightarrow\mathrm{x}+\mathrm{y}+\mathrm{2}\sqrt{\mathrm{xy}}\leqslant\mathrm{xy}+\mathrm{x}+\mathrm{y}+\mathrm{1}\Leftrightarrow\mathrm{xy}+\mathrm{1}\geqslant\mathrm{2}\sqrt{\mathrm{xy}},\mathrm{AM}−\mathrm{GM} \\ $$$$\Rightarrow\forall\left(\mathrm{x},\mathrm{y}\right)\in\mathbb{R}_{+} \sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}\leqslant\sqrt{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{y}+\mathrm{1}\right)} \\ $$$$\Rightarrow\forall\left(\mathrm{a},\mathrm{b}\right)\in\left[\mathrm{1},\infty\left[^{\mathrm{2}} \right.\right.…
Question Number 195126 by Erico last updated on 25/Jul/23 $$\mathrm{Soit}\:{f}_{{n}} \left({x}\right)=\mathrm{2}^{{n}+\mathrm{1}} \left[\frac{\frac{\mathrm{1}}{\mathrm{2}^{{n}} }{cotan}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)−{cotanx}}{{sin}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)}\right] \\ $$$${Calculer}\:\underset{{x}\rightarrow\mathrm{0}} {{lim}f}_{{n}} \left({x}\right)\:{et}\:\underset{{n}\rightarrow+\infty} {{lim}}\:\frac{{f}_{{n}} \left({x}\right)}{\mathrm{2}^{\mathrm{2}{n}+\mathrm{2}} } \\ $$ Answered…
Question Number 195121 by mathlove last updated on 25/Jul/23 $${x}^{\mathrm{2}} −{x}−\mathrm{1}=\mathrm{0} \\ $$$${x}^{\mathrm{8}} +\mathrm{2}{x}^{\mathrm{7}} −\mathrm{47}{x}=? \\ $$ Answered by qaz last updated on 25/Jul/23 $${x}^{\mathrm{8}}…
Question Number 195120 by sonukgindia last updated on 25/Jul/23 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 195154 by cortano12 last updated on 25/Jul/23 $$\:\:\:\:\underset{{x}\rightarrow\mathrm{2}\pi} {\mathrm{lim}}\:\left(\frac{\mathrm{tan}\:\left(\pi\:\mathrm{cos}\:{x}\right)}{{x}^{\mathrm{2}} \left({x}−\mathrm{5}\pi\right)+\mathrm{4}\pi^{\mathrm{2}} \left(\mathrm{2}{x}−\pi\right)}\right)=? \\ $$$$ \\ $$ Answered by dimentri last updated on 25/Jul/23 $$\:\:\:\underbrace{\Subset}…
Question Number 195122 by Rupesh123 last updated on 25/Jul/23 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 195148 by horsebrand11 last updated on 25/Jul/23 $$\:\:\begin{array}{|c|}{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}+\mathrm{x}\:\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}=?}\\\hline\end{array} \\ $$ Answered by Erico last updated on 25/Jul/23 $$\frac{\mathrm{1}+\mathrm{xsinx}−\mathrm{cosx}}{\mathrm{sin}^{\mathrm{2}} \mathrm{x}}=\frac{\mathrm{1}−\mathrm{cosx}}{\mathrm{sin}^{\mathrm{2}} \mathrm{x}}+\frac{\mathrm{x}}{\mathrm{sinx}} \\…
Question Number 195118 by York12 last updated on 25/Jul/23 $${a},{b},{c}>\mathrm{0}\:\&\:\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}=\mathrm{3} \\ $$$${prove}\:{that} \\ $$$$\frac{{a}}{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }+\frac{{b}}{{a}^{\mathrm{2}} +{c}^{\mathrm{2}} }+\frac{{c}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\geqslant\frac{\mathrm{3}}{\mathrm{2}}\left(\frac{{a}+{b}+{c}}{{ab}+{bc}+{ac}}\right)^{\mathrm{2}} \\ $$ Commented by York12…
Question Number 195175 by valdirmd last updated on 25/Jul/23 Answered by Rasheed.Sindhi last updated on 26/Jul/23 $${x}^{\mathrm{4}} +{x}^{\mathrm{2}} =\mathrm{11}/\mathrm{5}\:,\:\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}\right)^{\mathrm{1}/\mathrm{3}} =? \\ $$$$\blacktriangleright{Let}\:{y}=\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}\right)^{\mathrm{1}/\mathrm{3}} \\…