Question Number 21142 by oyshi last updated on 14/Sep/17 $${if}\:{A}+{B}=\frac{\pi}{\mathrm{4}} \\ $$$${so}\:{proof}\:\left(\mathrm{1}+\mathrm{tan}\:{A}\right)\left(\mathrm{1}+\mathrm{tan}\:{B}\right)=\mathrm{2} \\ $$ Commented by dioph last updated on 14/Sep/17 $$\mathrm{tan}\:{A}+{B}\:=\:\mathrm{1} \\ $$$$\frac{\mathrm{tan}\:{A}\:+\:\mathrm{tan}\:{B}}{\mathrm{1}\:−\:\mathrm{tan}\:{A}\:\mathrm{tan}\:{B}}\:=\:\mathrm{1} \\…
Question Number 21141 by oyshi last updated on 14/Sep/17 $${if}\:\mathrm{sin}\:\alpha\mathrm{sin}\:\beta−\mathrm{cos}\:\alpha\mathrm{cos}\:\beta+\mathrm{1}=\mathrm{0}\: \\ $$$${so}\:{proof}\:{that}\:\mathrm{1}+\mathrm{cot}\:\alpha\mathrm{tan}\:\beta=\mathrm{0} \\ $$ Answered by $@ty@m last updated on 15/Sep/17 $$\mathrm{sin}\:\alpha\mathrm{sin}\:\beta−\mathrm{cos}\:\alpha\mathrm{cos}\:\beta+\mathrm{1}=\mathrm{0}\: \\ $$$$\mathrm{cos}\:\alpha\mathrm{cos}\:\beta−\mathrm{sin}\:\alpha\mathrm{sin}\:\beta=\mathrm{1} \\…
Question Number 21140 by oyshi last updated on 14/Sep/17 $${if}\:\mathrm{tan}\:\beta=\frac{\mathrm{2sin}\:\alpha\mathrm{sin}\:\gamma}{\mathrm{sin}\:\left(\alpha+\gamma\right)} \\ $$$${so}\:{proof}\:\mathrm{cot}\:\gamma+\mathrm{cot}\:\alpha=\mathrm{2cot}\:\beta \\ $$ Commented by $@ty@m last updated on 15/Sep/17 $${Step}\:\mathrm{1}.\:{Take}\:{reciprocal}\:{of}\:{both}\:{sides}. \\ $$$${Step}\:\mathrm{2}.\:{Multiply}\:{both}\:{sides}\:{by}\:\mathrm{2} \\…
Question Number 152208 by SOMEDAVONG last updated on 26/Aug/21 $$\mathrm{1}.\mathrm{for}\:\forall\mathrm{x}>\mathrm{0}.\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{m}\:\mathrm{to}\: \\ $$$$\mathrm{1}+\mathrm{log}_{\mathrm{5}} \left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\geqslant\mathrm{log}_{\mathrm{5}} \left(\mathrm{mx}^{\mathrm{2}} +\mathrm{4x}+\mathrm{m}\right)\:\mathrm{verify}\:\forall\mathrm{x}. \\ $$ Answered by Rasheed.Sindhi last updated on 26/Aug/21…
Question Number 86675 by M±th+et£s last updated on 30/Mar/20 $$\underset{{x}\rightarrow\infty} {{lim}}\sqrt[{{n}}]{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}^{{n}} \right)^{{n}} {dx}}=? \\ $$ Commented by M±th+et£s last updated on 31/Mar/20 $$\underset{{n}\rightarrow\infty}…
Question Number 152211 by nitu last updated on 26/Aug/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} } \\ $$$$\mathrm{should}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{be}\:\infty\:\mathrm{or}\:\mathrm{is}\:\mathrm{it}\:\mathrm{DNE}. \\ $$$$\mathrm{my}\:\mathrm{main}\:\mathrm{question}\:\mathrm{is}, \\ $$$$\mathrm{when}, \\ $$$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\infty\: \\ $$$$\mathrm{does}\:\mathrm{it}\:\mathrm{not}\:\mathrm{exist}?\:\mathrm{is}\:\mathrm{it}\:\mathrm{DNE}? \\ $$…
Question Number 86672 by M±th+et£s last updated on 30/Mar/20 $${show}\:{proofs}\:{by}\:{induction},{that} \\ $$$$\frac{{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +….+{x}_{{n}} }{{n}}\geqslant\left({x}_{\mathrm{1}} {x}_{\mathrm{2}} ….{x}_{{n}} \right)^{\frac{\mathrm{1}}{{n}}} \\ $$$$\forall{n}=\mathrm{2}^{{k}} ,{k}>\mathrm{1}\:{and}\:\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,{x}_{\mathrm{3}} ,…..{x}_{{n}} \right)>\mathrm{0}.…
Question Number 21137 by Tinkutara last updated on 14/Sep/17 $$\mathrm{If}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mid{z}+\mathrm{1}+{i}\mid\:+\:\mid{z}−\mathrm{1}−{i}\mid\:+\:\mid\mathrm{2}\:−\:{z}\mid\:+\:\mid\mathrm{3}\:−\:{z}\mid\:\mathrm{is} \\ $$$${k}\:\mathrm{then}\:\left({k}\:−\:\mathrm{8}\right)\:\mathrm{equals} \\ $$ Answered by Tinkutara last updated on 15/Sep/17 $$\mid{z}_{\mathrm{1}} \mid+\mid{z}_{\mathrm{2}}…
Question Number 152204 by cherokeesay last updated on 26/Aug/21 Commented by Paradoxical last updated on 26/Aug/21 Commented by cherokeesay last updated on 26/Aug/21 $${Nice}\:! \\…
Question Number 86671 by M±th+et£s last updated on 30/Mar/20 $$\int{sin}\left({x}\right)\:{arcsin}\left({x}\right) \\ $$ Answered by Rio Michael last updated on 30/Mar/20 $$\mathrm{let}\:\mathrm{me}\:\mathrm{give}\:\mathrm{a}\:\mathrm{try}. \\ $$$$\:\int\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\left({x}\right)\:{dx} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{taylor}\:\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\:\mathrm{sin}\:\left({x}\right)\:\mathrm{arcsin}\:\left({x}\right)\:\mathrm{centred}\:\mathrm{at}\:\mathrm{0}…