Question Number 135679 by liberty last updated on 15/Mar/21 $$\left(\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\right)\left(\mathrm{2tan}\:{x}+\frac{\mathrm{1}}{\mathrm{cos}\:{x}}\right)+\mathrm{2}\:=\:\mathrm{0} \\ $$ Answered by EDWIN88 last updated on 15/Mar/21 $$\mathrm{Let}\:\mathrm{tan}\:\frac{\mathrm{x}}{\mathrm{2}}\:=\:\mathrm{t}\:\rightarrow\begin{cases}{\mathrm{cos}\:\mathrm{x}=\frac{\mathrm{1}−\mathrm{t}}{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }}\\{\mathrm{sin}\:\mathrm{x}=\frac{\mathrm{2t}}{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }}\\{\mathrm{tan}\:\mathrm{x}=\frac{\mathrm{2t}}{\mathrm{1}−\mathrm{t}^{\mathrm{2}} }}\end{cases} \\ $$$$\Leftrightarrow\:\left(\frac{\mathrm{1}−\mathrm{t}^{\mathrm{2}}…
Question Number 135673 by liberty last updated on 14/Mar/21 $$\int\:\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}\: \\ $$ Answered by Olaf last updated on 15/Mar/21 $$\Omega\:=\:\int\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}}…
Question Number 70138 by Scientist0000001 last updated on 01/Oct/19 $${prove}\:{that}\:\:\:{e}^{{i}\theta} ={e}^{{i}\left(\theta+\mathrm{2}{k}\Pi\right)} \:\:{given}\:{that}\:{k}=\mathrm{0},\pm\mathrm{1},\pm\mathrm{2}… \\ $$ Answered by mind is power last updated on 01/Oct/19 $${e}^{{ix}} ={cos}\left({x}\right)+{isin}\left({x}\right)…
Question Number 135672 by Raxreedoroid last updated on 14/Mar/21 $$\mathrm{Write}\:\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{2}} {\sum}}{kx}^{{k}−\mathrm{1}} \:\mathrm{in}\:\mathrm{terms}\:{n} \\ $$ Answered by Olaf last updated on 15/Mar/21 $${a}^{{n}} −{b}^{{n}} \:=\:\left({a}−{b}\right)\left({a}^{{n}−\mathrm{1}}…
Question Number 4600 by Yozzii last updated on 11/Feb/16 $${Find}\:{the}\:{value}\:{of}\:{S}\:{where}\:{S}=\underset{{r}=\mathrm{1}} {\overset{\mathrm{89}} {\sum}}{sin}^{\mathrm{6}} {r}°. \\ $$ Commented by FilupSmith last updated on 11/Feb/16 $$\mathrm{Don}'\mathrm{t}\:\mathrm{you}\:\mathrm{use}\:\mathrm{the}\:\mathrm{series}\:\mathrm{representation} \\ $$$$\mathrm{for}\:\mathrm{sine}?…
Question Number 70135 by petrochengula last updated on 01/Oct/19 $${sophie}−{Germain}\:{identity} \\ $$$${a}^{\mathrm{4}} +\mathrm{4}{b}^{\mathrm{4}} =\left(\left({a}+{b}\right)^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\left(\left({a}−{b}\right)^{\mathrm{2}} +{b}^{\mathrm{2}} \right) \\ $$ Commented by mathmax by abdo…
Question Number 70132 by RAKESH MANDA last updated on 01/Oct/19 $$\underset{{n}=\mathrm{1}} {\overset{\mathrm{3050}} {\sum}}\:{i}^{{n}} \\ $$ Commented by mathmax by abdo last updated on 01/Oct/19 $${S}\:=\sum_{{n}=\mathrm{0}}…
Question Number 4597 by FilupSmith last updated on 10/Feb/16 $${a}_{{n}} =\sqrt{{a}_{{n}−\mathrm{1}} } \\ $$$${a}_{\mathrm{1}} ={c} \\ $$$$ \\ $$$$\mathrm{Solve}: \\ $$$${S}\:=\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \\ $$ Commented…
Question Number 4596 by Yozzii last updated on 10/Feb/16 $${Use}\:{the}\:{definition}\:{of}\:{the}\:{limit}\:{of}\:{a}\:{function} \\ $$$${to}\:{prove}\:{that}\:\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\frac{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} }{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\mathrm{0}. \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 4594 by alimyao last updated on 10/Feb/16 $$\int\mathrm{4}{x}^{\mathrm{2}} \:−\:\frac{\mathrm{5}}{\mathrm{2}{x}^{−\mathrm{2}} }\:+\:\mathrm{4}\:\:{dx},\:{x}\neq\mathrm{0} \\ $$$$ \\ $$ Answered by FilupSmith last updated on 10/Feb/16 $$=\mathrm{4}\int{x}^{\mathrm{2}} {dx}−\frac{\mathrm{5}}{\mathrm{2}}\int{x}^{\mathrm{2}}…