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Question Number 195365 by Mr.D.N. last updated on 31/Jul/23 $$\:\: \\ $$$$\:\underset{\mathrm{x}\rightarrow\mathrm{y}} {\overset{\mathrm{lim}} {\:}}\:\frac{\mathrm{tan}\:\mathrm{x}−\mathrm{tany}}{\mathrm{x}−\mathrm{y}} \\ $$$$ \\ $$ Answered by cortano12 last updated on 31/Jul/23…
Question Number 195364 by Rodier97 last updated on 01/Aug/23 $$ \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{natural}\:\mathrm{number}\:{n},\: \\ $$$$\mathrm{the}\:\mathrm{natural}\:\mathrm{number}\:\left(\mathrm{3}−\sqrt{\mathrm{5}}\right)^{{n}} +\left(\mathrm{3}+\sqrt{\mathrm{5}}\right)^{{n}} \:\mathrm{is}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{2}^{{n}} . \\ $$ Answered by Frix last…
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Question Number 195325 by mathlove last updated on 30/Jul/23 $${prove}\:{that} \\ $$$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{{tan}\left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{1}}{{x}−\frac{\pi}{\mathrm{2}}}=\mathrm{1} \\ $$ Answered by BaliramKumar last updated on 30/Jul/23 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\frac{\mathrm{d}}{\mathrm{dx}}\left({tan}\left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{1}\right)}{\frac{\mathrm{d}}{\mathrm{dx}}\left({x}−\frac{\pi}{\mathrm{2}}\right)}\:=\:\frac{\mathrm{sec}^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)\centerdot\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{1}}…
Question Number 195320 by Erico last updated on 30/Jul/23 $$\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\:+\infty} {t}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} {tdt} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{I}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{1}−{e}^{−\mathrm{2}\pi} }\:\:\underset{\:\mathrm{0}} {\int}^{\:\pi} {e}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} {t}\:{dt} \\ $$$$\mathrm{and}\:\:\mathrm{I}_{\mathrm{n}}…
Question Number 195322 by Rupesh123 last updated on 30/Jul/23 Answered by HeferH last updated on 31/Jul/23 $${say}\:{k}\:{is}\:{the}\:{incenter}\:{length} \\ $$$$\:{k}\:=\:\frac{{A}}{{s}} \\ $$$$\:{s}\:=\:\frac{{AB}+{AC}+{BC}}{\mathrm{2}} \\ $$$$\:{A}\:=\:\frac{\mathrm{3}{k}\centerdot{BC}}{\mathrm{2}} \\ $$$$\:\mathrm{1}\:=\:\frac{\mathrm{3}{BC}}{{AB}+{AC}+{BC}}…
Question Number 195331 by Shlock last updated on 30/Jul/23 Answered by mr W last updated on 06/Aug/23 $${say}\:{a}={pc},\:{b}={qc} \\ $$$$\left({p}+{q}+\mathrm{1}\right)\left(\frac{\mathrm{1}}{{p}}+\frac{\mathrm{1}}{{q}}+\mathrm{1}\right)=\mathrm{10} \\ $$$$\frac{{a}+{b}}{{c}}={p}+{q} \\ $$$$\left({p}+{q}\right)_{{max}} =?…
Question Number 195315 by cortano12 last updated on 30/Jul/23 $$\:\:\:\:\:\:\Subset \\ $$ Answered by horsebrand11 last updated on 30/Jul/23 $$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:^{\mathrm{2}} \mathrm{a}−\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}−\mathrm{tan}\:^{\mathrm{2}} \mathrm{a}+\mathrm{tan}\:^{\mathrm{4}} \mathrm{a}\:\mathrm{tan}\:^{\mathrm{2}}…