Question Number 13658 by Tinkutara last updated on 22/May/17 $$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{cos}^{\mathrm{2}} {x}\:+\:\mathrm{cos}^{\mathrm{2}} \mathrm{3}{x}\:+\:\mathrm{cos}^{\mathrm{2}} \mathrm{5}{x}\:+\:…\:\mathrm{to}\:{n}\:\mathrm{terms} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\left[{n}\:+\:\frac{\mathrm{sin4}{nx}}{\mathrm{2sin2}{x}}\right] \\ $$ Answered by ajfour last updated on…
Question Number 144727 by imjagoll last updated on 28/Jun/21 $$\:\:\mathrm{Given}\:\begin{cases}{\mathrm{m}=\mathrm{cos}\:\theta−\mathrm{sin}\:\theta}\\{\mathrm{n}=\mathrm{cos}\:\theta+\mathrm{sin}\:\theta}\end{cases} \\ $$$$\:\:\mathrm{then}\:\sqrt{\frac{\mathrm{m}}{\mathrm{n}}}\:+\sqrt{\frac{\mathrm{n}}{\mathrm{m}}}\:=\:? \\ $$ Answered by liberty last updated on 28/Jun/21 $$\:\Leftrightarrow\sqrt{\frac{\mathrm{m}}{\mathrm{n}}}+\sqrt{\frac{\mathrm{n}}{\mathrm{m}}}=\:\frac{\mathrm{m}+\mathrm{n}}{\:\sqrt{\mathrm{mn}}} \\ $$$$\begin{cases}{\mathrm{m}^{\mathrm{2}} =\mathrm{cos}\:^{\mathrm{2}}…
Question Number 13654 by Tinkutara last updated on 22/May/17 $$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\mathrm{sin}\theta\:+\:\mathrm{sin}\left(\frac{{n}\:−\:\mathrm{4}}{{n}\:−\:\mathrm{2}}\right)\theta\:+\:\mathrm{sin}\left(\frac{{n}\:−\:\mathrm{6}}{{n}\:−\:\mathrm{2}}\right)\theta\:+\:…\:{n}\:\mathrm{terms} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{sin}\left(\frac{{n}\theta}{\mathrm{2}\:−\:{n}}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{cos}\left(\frac{\mathrm{2}{n}\theta}{\mathrm{2}\:−\:{n}}\right) \\ $$$$\left(\mathrm{3}\right)\:\mathrm{tan}{n}\theta \\ $$$$\left(\mathrm{4}\right)\:\mathrm{cot}{n}\theta \\ $$ Answered…
Question Number 144614 by nadovic last updated on 27/Jun/21 $$\overset{} {\:}\mathrm{Given}\:\mathrm{that}\:{x}\:=\:\mathrm{tan}\:\mathrm{23}°,\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\:\mathrm{of}\:\:\mathrm{cos}\:\mathrm{16}°\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{x}\underset{} {.} \\ $$ Answered by qaz last updated on 27/Jun/21 $$\mathrm{x}=\mathrm{tan}\:\mathrm{23}°=\frac{\mathrm{tan}\:\mathrm{30}°−\mathrm{tan}\:\mathrm{7}°}{\mathrm{1}+\mathrm{tan}\:\mathrm{30}°\centerdot\mathrm{tan}\:\mathrm{7}°} \\…
Question Number 79075 by mathocean1 last updated on 22/Jan/20 $$\mathrm{Hello}\: \\ $$$$\mathrm{solve}\:\mathrm{in}\:\left[\mathrm{0};\mathrm{2}\pi\right]\:\mathrm{tan2}{x}\geqslant\sqrt{\mathrm{3}} \\ $$ Answered by mr W last updated on 22/Jan/20 $$\mathrm{tan}\:\mathrm{2}{x}\geqslant\sqrt{\mathrm{3}} \\ $$$$\Rightarrow{k}\pi+\frac{\pi}{\mathrm{3}}\leqslant\mathrm{2}{x}<{k}\pi+\frac{\pi}{\mathrm{2}}…
Question Number 79062 by mr W last updated on 22/Jan/20 $${if}\:{a}\:{is}\:{a}\:{rational}\:{number}\:{with}\:\mid{a}\mid\leqslant\mathrm{1}, \\ $$$${prove}\:{that}\:\mathrm{cos}\:\left({n}\:\mathrm{cos}^{−\mathrm{1}} \left({a}\right)\right)\:{is}\:{also}\:{a} \\ $$$${rational}\:{number}.\:\left({n}\in\mathbb{N}\right) \\ $$ Answered by mind is power last updated…
Question Number 78946 by mathocean1 last updated on 21/Jan/20 $$\mathrm{solve} \\ $$$$\mathrm{cos}{x}−\sqrt{\mathrm{3}}{sinx}=\mathrm{1} \\ $$ Commented by msup trace by abdo last updated on 21/Jan/20 $${e}\:\Leftrightarrow\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}{cosx}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{sinx}\right)=\mathrm{1}\:\Rightarrow…
Question Number 78944 by mathocean1 last updated on 21/Jan/20 $$\mathrm{Hello}\:\mathrm{sirs}\:\mathrm{i}\:\mathrm{need}\:\mathrm{your}\:\mathrm{help}\:\mathrm{to}\:\mathrm{solve} \\ $$$$\mathrm{tan2}{x}\geqslant\sqrt{\mathrm{3}}\:\mathrm{in}\:\left[\mathrm{0};\mathrm{2}\pi\right]. \\ $$$$\mathrm{i}\:\mathrm{want}\:\mathrm{that}\:\mathrm{you}\:\mathrm{explain}\:\mathrm{me}\:\mathrm{if}\: \\ $$$$\mathrm{possible}\:\mathrm{how}\:\mathrm{we}\:\mathrm{make}\:\mathrm{graphic}\:\mathrm{to} \\ $$$$\mathrm{determinate}. \\ $$ Commented by mathocean1 last updated…
Question Number 144483 by alcohol last updated on 25/Jun/21 $$\left({p}_{{n}} \right)=\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)\left(\mathrm{1}+\frac{\mathrm{2}}{{n}^{\mathrm{2}} }\right)…\left(\mathrm{1}+\frac{{n}}{{n}^{\mathrm{2}} }\right) \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{6}}{n}\left(\mathrm{2}{n}+\mathrm{1}\right)\left({n}+\mathrm{1}\right) \\ $$$${show}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)−\frac{\mathrm{1}}{\mathrm{12}{n}^{\mathrm{2}} }\left(\mathrm{2}{n}+\mathrm{1}\right)\left({n}+\mathrm{1}\right)<{ln}\left({p}_{{n}} \right)<\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)…
Question Number 13403 by Tinkutara last updated on 19/May/17 $$\mathrm{If}\:\mathrm{tan}\:\left({A}\:−\:{B}\right)\:=\:\mathrm{1},\:\mathrm{sec}\:\left({A}\:+\:{B}\right)\:=\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\:, \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive} \\ $$$$\mathrm{value}\:\mathrm{of}\:{B}\:\mathrm{is}\:\frac{\mathrm{19}\pi}{\mathrm{24}}\:. \\ $$ Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 19/May/17 $${excuse}\:{me}\:,{but}\:{i}\:{think}\:{your}\:{answer}\:\left(\frac{\mathrm{19}\pi}{\mathrm{24}}\right)\:{is}\:{not}\:{true}. \\…