Question Number 129807 by bramlexs22 last updated on 19/Jan/21 $$\:\mathrm{8}.\left(\mathrm{1}+\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)\left(\mathrm{1}+\mathrm{sin}\frac{\mathrm{3}\pi}{\mathrm{8}}\right)\left(\mathrm{1}−\mathrm{sin}\:\frac{\mathrm{5}\pi}{\mathrm{8}}\right)\left(\mathrm{1}−\mathrm{sin}\:\frac{\mathrm{7}\pi}{\mathrm{8}}\right)=?\: \\ $$ Answered by EDWIN88 last updated on 19/Jan/21 $$\:\mathrm{sin}\:\mathrm{x}\:=\:\mathrm{sin}\:\left(\pi−\mathrm{x}\right) \\ $$$$\:\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{8}}=\mathrm{sin}\:\frac{\mathrm{5}\pi}{\mathrm{8}}\:\mathrm{and}\:\mathrm{sin}\:\frac{\mathrm{7}\pi}{\mathrm{8}}=\mathrm{sin}\:\frac{\pi}{\mathrm{8}} \\ $$$$\Leftrightarrow\:\mathrm{8}\left(\mathrm{1}+\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\:\mathrm{8}}\right)\left(\mathrm{1}+\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{8}}\right)\left(\mathrm{1}−\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{8}}\right)= \\…
Question Number 129801 by bramlexs22 last updated on 19/Jan/21 $$\:\mathrm{cot}\:\mathrm{15}°\:\mathrm{cot}\:\mathrm{16}°\:\mathrm{cot}\:\mathrm{17}°…\:\mathrm{cot}\:\mathrm{75}°= \\ $$ Answered by EDWIN88 last updated on 19/Jan/21 $$=\mathrm{1} \\ $$$$\:\left[\:\mathrm{cot}\:\mathrm{15}°=\mathrm{tan}\:\mathrm{75}°\:\mathrm{etc}\:\right]\: \\ $$ Terms…
Question Number 129719 by bramlexs22 last updated on 18/Jan/21 $$\:\:\mathrm{cos}\:\left(\pi\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{3}}\right)\right)=\mathrm{sin}\:\left(\pi\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{3}}\right)\right) \\ $$$$\:\:\mathrm{x}\:=? \\ $$$$ \\ $$ Answered by liberty last updated on 18/Jan/21 $$\:\mathrm{cos}\:\left(\pi\mathrm{sin}\:\frac{\mathrm{x}}{\mathrm{3}}\right)=\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}−\pi\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{3}}\right) \\…
Question Number 129712 by liberty last updated on 18/Jan/21 Commented by bemath last updated on 18/Jan/21 $$\mathrm{edit}\:\mathrm{the}\:\mathrm{last}\:\mathrm{line}\equiv\:\frac{\mathrm{3}−\mathrm{0}+\frac{\mathrm{3}}{\mathrm{4}}−\frac{\mathrm{3}}{\mathrm{8}}}{\mathrm{1}−\mathrm{0}−\frac{\mathrm{3}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{8}}}=\mathrm{9} \\ $$ Commented by liberty last updated on…
Question Number 64173 by Eminem last updated on 14/Jul/19 $${show}\:{tbat}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{1}−\frac{{sin}^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}{n}}\right)}{{sin}^{\mathrm{2}} \left(\frac{\mathrm{2}{k}−\mathrm{1}}{\mathrm{4}}\pi\right)}\right)={cos}\left(\theta\right) \\ $$ Commented by Prithwish sen last updated on 15/Jul/19 $$\mathrm{Is}'\mathrm{nt}\:\mathrm{it}\:\mathrm{cos}^{\mathrm{n}}…
Question Number 129360 by bramlexs22 last updated on 15/Jan/21 $$\begin{cases}{\mathrm{sin}\:{a}\:+\:\mathrm{sin}\:{b}\:=\:\frac{\mathrm{1}}{\mathrm{3}}}\\{\mathrm{cos}\:{a}+\mathrm{cos}\:{b}\:=\:\frac{\mathrm{5}}{\mathrm{3}}}\end{cases} \\ $$$$\:\:\:\:\mathrm{cos}^{\mathrm{2}} \:\left({a}+{b}\right)\:=?\: \\ $$ Commented by bramlexs22 last updated on 15/Jan/21 $$\:\mathrm{nice}\:\mathrm{trigonometry} \\ $$…
Question Number 63747 by Tony Lin last updated on 08/Jul/19 $${What}\:{does}\:{sin}^{−\mathrm{2}} {x}\:{mean}? \\ $$ Commented by MJS last updated on 08/Jul/19 $$\mathrm{that}'\mathrm{s}\:\mathrm{the}\:\mathrm{question}… \\ $$$$\mathrm{we}'\mathrm{re}\:\mathrm{not}\:\mathrm{consequent}\:\mathrm{in}\:\mathrm{these}\:\mathrm{things} \\…
Question Number 63703 by kaivan.ahmadi last updated on 07/Jul/19 $${sin}^{\mathrm{3}} {x}+{cos}^{\mathrm{3}} {x}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{sin}\mathrm{2}{x}\:\::{x}\in\left[\mathrm{0},\mathrm{2}\pi\right]. \\ $$$${find}\:\:{x} \\ $$ Commented by Prithwish sen last updated on 08/Jul/19 $$\left(\mathrm{sinx}\:+\:\mathrm{cosx}\:\right)\:\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin2x}\right)\:−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin2x}\right)=\mathrm{0}…
Question Number 129234 by bemath last updated on 14/Jan/21 $$\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{47}°+\mathrm{cos}^{\mathrm{2}} \:\:\mathrm{73}°+\mathrm{cos}\:\mathrm{47}°\mathrm{cos}\:\mathrm{73}°+\frac{\mathrm{1}}{\mathrm{2}}=? \\ $$ Answered by liberty last updated on 14/Jan/21 $$\left(\mathrm{cos}\:\mathrm{73}°+\mathrm{cos}\:\mathrm{47}°\right)^{\mathrm{2}} −\mathrm{cos}\:\mathrm{73}°\mathrm{cos}\:\mathrm{47}°+\frac{\mathrm{1}}{\mathrm{2}}= \\ $$$$\left(\mathrm{2cos}\:\mathrm{60}°\mathrm{cos}\:\mathrm{13}°\right)^{\mathrm{2}}…
Question Number 129233 by MrJoe last updated on 14/Jan/21 $${prove}\:{that} \\ $$$${sin}\:\mathrm{3}{x}=\mathrm{2}{cos}\left(\frac{\mathrm{3}{x}}{\mathrm{2}}\right){sin}\left(\frac{\mathrm{3}{x}}{\mathrm{2}}\right) \\ $$ Commented by MrJoe last updated on 15/Jan/21 $${then}\: \\ $$$${if}\:{sin}\left(\mathrm{3}{x}\right)={sin}\left[\mathrm{2}\left(\frac{\mathrm{3}{x}}{\mathrm{2}}\right)\right] \\…