Question Number 146044 by mathdanisur last updated on 10/Jul/21 $${Simplify}: \\ $$$$\frac{{sin}^{\mathrm{3}} \alpha}{\mathrm{1}-{cos}\alpha}\:+\:\frac{{cos}^{\mathrm{3}} \alpha}{{sin}\alpha+\mathrm{1}}\:=\:? \\ $$ Answered by liberty last updated on 10/Jul/21 $$\:\Rightarrow\:\frac{\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \alpha\right)\mathrm{sin}\:\alpha}{\mathrm{1}−\mathrm{cos}\:\alpha}\:+\:\frac{\left(\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}}…
Question Number 146046 by mathdanisur last updated on 10/Jul/21 $${f}\left({x}\right)\:=\:{x}^{\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)} \:\Rightarrow\:{f}\:^{'} \left({x}\right)\:=\:? \\ $$ Answered by liberty last updated on 10/Jul/21 $$\:\mathrm{ln}\:{f}\left({x}\right)=\mathrm{sin}\:{x}.\:\mathrm{ln}\:\left({x}\right) \\ $$$$\Leftrightarrow\:\frac{{f}\:'\left({x}\right)}{{f}\left({x}\right)}=\:\mathrm{cos}\:{x}.\mathrm{ln}\:\left({x}\right)+\frac{\mathrm{sin}\:{x}}{{x}} \\…
Question Number 80503 by Rio Michael last updated on 03/Feb/20 $$\mathrm{prove}\:\mathrm{that}\:\mathrm{they}\:\mathrm{are}\:\mathrm{infinitely}\:\mathrm{many} \\ $$$$\mathrm{primes} \\ $$ Answered by MJS last updated on 03/Feb/20 $$\mathrm{if}\:\mathrm{there}\:\mathrm{are}\:{n}\:\mathrm{primes},\:{p}_{\mathrm{1}} ,\:{p}_{\mathrm{2}} ,\:…\:{p}_{{n}}…
Question Number 80493 by TawaTawa last updated on 03/Feb/20 $$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{a},\:\mathrm{b}\:\mathrm{and}\:\mathrm{c} \\ $$$$\:\:\:\:\:\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\:…..\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{abc}\:\:\:=\:\:\:−\:\frac{\mathrm{1}}{\mathrm{4}}\:\:\:\:\:……\:\left(\mathrm{iii}\right) \\ $$$$\:\:\:\:\:\:\mathrm{ab}\:+\:\mathrm{ac}\:+\:\mathrm{bc}\:\:\:=\:\:\frac{\mathrm{3}}{\mathrm{2}}\:\:\:\:\:\:\:……\:\left(\mathrm{iv}\right) \\ $$ Commented by mr W last updated on…
Question Number 146013 by mathdanisur last updated on 10/Jul/21 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 146009 by iloveisrael last updated on 10/Jul/21 $$\:\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{12}}+\frac{\mathrm{1}}{\mathrm{24}}+…+\frac{\mathrm{1}}{\mathrm{2n}\left(\mathrm{n}+\mathrm{1}\right)}=? \\ $$ Answered by puissant last updated on 10/Jul/21 $$=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{2k}\left(\mathrm{k}+\mathrm{1}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}}…
Question Number 146001 by puissant last updated on 10/Jul/21 $$\mathrm{Soit}\:\mathrm{p}\in\mathrm{End}\left(\mathrm{E}\right).\:\mathrm{on}\:\mathrm{pose}\:\mathrm{q}=\mathrm{id}_{\mathrm{E}} −\mathrm{p} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{montrer}\:\mathrm{que}\:\mathrm{p}\:\mathrm{est}\:\mathrm{un}\:\mathrm{projecteur}\:\mathrm{si}\:\mathrm{et}\: \\ $$$$\mathrm{seulement}\:\mathrm{si}\:\mathrm{q}\:\mathrm{est}\:\mathrm{un}\:\mathrm{projecteur}.. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{on}\:\mathrm{suppose}\:\mathrm{que}\:\mathrm{p}\:\mathrm{est}\:\mathrm{un}\:\mathrm{projecteur}\:\mathrm{et}\:\mathrm{on} \\ $$$$\mathrm{considere}\:\mathrm{L}=\left\{\mathrm{f}\in\mathrm{End}\left(\mathrm{E}\right)/\exists\mathrm{u}\in\mathrm{End}\left(\mathrm{E}\right),\mathrm{f}=\mathrm{u}\circ\mathrm{p}\right\} \\ $$$$\mathrm{et}\:\mathrm{M}=\left\{\mathrm{g}\in\mathrm{End}\left(\mathrm{E}\right)/\exists\mathrm{v}\in\mathrm{End}\left(\mathrm{E}\right),\:\mathrm{g}=\mathrm{v}\circ\mathrm{q}\right\}. \\ $$$$\mathrm{montrer}\:\mathrm{que}\:\mathrm{L}\:\mathrm{et}\:\mathrm{M}\:\mathrm{sont}\:\mathrm{des}\:\mathrm{sous}\:\mathrm{espaces}\: \\ $$$$\mathrm{vectoriels}\:\mathrm{supplementaires}\:\mathrm{de}\:\mathrm{End}\left(\mathrm{E}\right)..…
Question Number 146004 by puissant last updated on 10/Jul/21 $$\mathrm{F}\:\mathrm{et}\:\mathrm{G}\:\mathrm{deux}\:\mathrm{sous}\:\mathrm{espaces}\:\mathrm{vectoriels}\:\mathrm{de}\:\mathrm{E} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{montrer}\:\mathrm{que}\:\left(\mathrm{F}\cap\mathrm{G}=\mathrm{F}+\mathrm{G}\right)\Leftrightarrow\left(\mathrm{F}=\mathrm{G}\right) \\ $$$$\left.\mathrm{b}\right)\:\mathrm{quand}\:\mathrm{dit}−\mathrm{on}\:\mathrm{que}\:\mathrm{les}\:\mathrm{deux}\:\mathrm{sous}\:\mathrm{espaces}\: \\ $$$$\mathrm{vectoriels}\:\mathrm{F}\:\mathrm{et}\:\mathrm{G}\:\mathrm{sont}\:\mathrm{supplementaires}? \\ $$ Answered by Olaf_Thorendsen last updated on 10/Jul/21…
Question Number 145996 by iloveisrael last updated on 10/Jul/21 $$\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\: \\ $$$$\:\mathrm{x}−\mathrm{2x}^{\mathrm{2}} +\mathrm{3x}^{\mathrm{3}} −\mathrm{4x}^{\mathrm{4}} +\mathrm{5x}^{\mathrm{5}} −\mathrm{6x}^{\mathrm{6}} +… \\ $$$$\mathrm{where}\:\mid\mathrm{x}\mid\:<\:\mathrm{1} \\ $$ Answered by mathmax by…
Question Number 80448 by mind is power last updated on 03/Feb/20 $${Hello}\:{All}\:{of}\:{You}\:{verry}\:{Nice}\:{Day},\:{God}\:{bless}\:{You}\:{love}\:{peace}\:{and}\: \\ $$$${happiness}\: \\ $$$${Solve}\:{for}\:\left({x},{y}\right)\in\mathbb{R}^{\mathrm{2}} \: \\ $$$$\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2}{x}+\mathrm{3}{y}+\mathrm{1}}\\{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} =\mathrm{4}{x}^{\mathrm{2}} +\mathrm{9}{y}^{\mathrm{2}} +\mathrm{12}{xy}+\mathrm{2}{x}^{\mathrm{2}}…