Question Number 101040 by Rio Michael last updated on 30/Jun/20 $$\:\mathrm{Express}\:\mathrm{2}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\mathrm{6}\theta\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\:\mathrm{sin}\:{A}\:−\:\mathrm{sin}\:{B} \\ $$$$\left({i}\right)\:\mathrm{using}\:\mathrm{that}\:\mathrm{result}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{2sin}\:\theta\left(\:\mathrm{cos}\:\mathrm{6}\theta\:+\:\mathrm{cos}\:\mathrm{4}\theta\:+\:\mathrm{cos}\:\mathrm{2}\theta\right)\:=\:\mathrm{sin}\:\mathrm{7}\theta−\mathrm{sin}\:\theta \\ $$$$\left({ii}\right)\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{result}\:\mathrm{cos}\:\left(\frac{\mathrm{12}\pi}{\mathrm{7}}\right)\:+\:\mathrm{cos}\:\left(\frac{\mathrm{8}\pi}{\mathrm{7}}\right)\:+\:\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{7}}\right)\:=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left({iii}\right)\:\mathrm{hence}\:\mathrm{find}\:\mathrm{a}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{to}\:\frac{\mathrm{sin7}\theta\:−\:\mathrm{sin}\:\theta}{\mathrm{cos}\:\mathrm{6}\theta\:+\:\mathrm{cos}\:\mathrm{4}\theta\:+\:\mathrm{cos}\:\mathrm{2}\theta}\:=\:\mathrm{1} \\ $$ Answered by maths mind last updated…
Question Number 100988 by bobhans last updated on 29/Jun/20 $$\mathrm{4sin}\:^{\mathrm{2}} {x}\:+\:\mathrm{sin}\:\mathrm{2}{x}\:=\:\mathrm{3}\: \\ $$$${find}\:{solution}\:{set}\:{on}\:{x}\in\left(\mathrm{0},\mathrm{2}\pi\right) \\ $$ Commented by Dwaipayan Shikari last updated on 29/Jun/20 $${sin}^{\mathrm{2}} {x}+\mathrm{2}{sinxcosx}+{cos}^{\mathrm{2}}…
Question Number 35362 by 26488679 last updated on 18/May/18 Commented by 26488679 last updated on 18/May/18 $${is}\:{arcsin}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\:{can}\:{be}\:\:{equal}\:{to}\:\mathrm{210}°? \\ $$ Commented by ajfour last updated on…
Question Number 100870 by bramlex last updated on 29/Jun/20 Commented by bemath last updated on 29/Jun/20 $$\pi−\mathrm{e}\:=\:\mathrm{0}.\mathrm{4233} \\ $$$$\left(\pi−\mathrm{e}\right)^{\mathrm{ln}\left(\mathrm{1}−\mathrm{2cos}\:^{\mathrm{2}} \mathrm{x}\right)} \:\geqslant\:\left(\pi−\mathrm{e}\right)^{\mathrm{0}} \\ $$$$\mathrm{case}\left(\mathrm{1}\right)\Rightarrow\mathrm{ln}\left(\mathrm{1}−\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{2x}\right)\right)\leqslant\:\mathrm{0} \\ $$$$\mathrm{ln}\left(−\mathrm{cos}\:\mathrm{2x}\right)\:\leqslant\:\mathrm{0}\:\Rightarrow−\mathrm{cos}\:\mathrm{2x}\:\leqslant\:\mathrm{1}…
Question Number 166360 by mathlove last updated on 19/Feb/22 $$\frac{\mathrm{sin}\:\mathrm{10}{x}}{{sin}\:\mathrm{2}{x}}−\frac{\mathrm{cos}\:\mathrm{10}{x}}{\mathrm{cos}\:\mathrm{2}{x}}=? \\ $$ Answered by som(math1967) last updated on 19/Feb/22 $$\frac{\boldsymbol{{sin}}\mathrm{10}\boldsymbol{{xcos}}\mathrm{2}\boldsymbol{{x}}−\boldsymbol{{cos}}\mathrm{10}\boldsymbol{{xsin}}\mathrm{2}\boldsymbol{{x}}}{\boldsymbol{{sin}}\mathrm{2}\boldsymbol{{xcos}}\mathrm{2}\boldsymbol{{x}}} \\ $$$$=\frac{\boldsymbol{{sin}}\left(\mathrm{10}\boldsymbol{{x}}−\mathrm{2}\boldsymbol{{x}}\right)}{\boldsymbol{{sin}}\mathrm{2}\boldsymbol{{xcos}}\mathrm{2}\boldsymbol{{x}}}=\frac{\mathrm{2}\boldsymbol{{sin}}\mathrm{8}\boldsymbol{{x}}}{\mathrm{2}\boldsymbol{{sin}}\mathrm{2}\boldsymbol{{xcos}}\mathrm{2}\boldsymbol{{x}}} \\ $$$$=\frac{\mathrm{4}\boldsymbol{{sin}}\mathrm{4}\boldsymbol{{xcos}}\mathrm{4}\boldsymbol{{x}}}{\boldsymbol{{sin}}\mathrm{4}\boldsymbol{{x}}}=\mathrm{4}\boldsymbol{{cos}}\mathrm{4}\boldsymbol{{x}} \\…
Question Number 166296 by mathlove last updated on 18/Feb/22 $$\left(\mathrm{cos}\:{x}\right)^{\mathrm{2022}} −\left(\mathrm{sin}\:{x}\right)^{\mathrm{2022}} =\mathrm{1} \\ $$$${x}=? \\ $$ Commented by mkam last updated on 18/Feb/22 $$\boldsymbol{{x}}\:=\:\boldsymbol{{n}\pi}\:\forall\:\boldsymbol{{n}}\:\in\:\boldsymbol{{Z}}\: \\…
Question Number 100695 by john santu last updated on 28/Jun/20 Commented by bobhans last updated on 28/Jun/20 $$\mathrm{sin}\:\mathrm{18}\:=\:\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}\:,\:\mathrm{sin}\:\mathrm{54}\:=\:\mathrm{cos}\:\mathrm{36}\:=\:\mathrm{1}−\mathrm{2sin}\:^{\mathrm{2}} \mathrm{18} \\ $$$$\mathrm{sin}\:\mathrm{54}\:=\:\mathrm{1}−\mathrm{2}\left(\frac{\mathrm{6}−\mathrm{2}\sqrt{\mathrm{5}}}{\mathrm{16}}\right)\:=\:\mathrm{1}−\left(\frac{\mathrm{3}−\sqrt{\mathrm{5}}}{\mathrm{4}}\right)=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{18}}\:−\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{54}}\:=\:\frac{\mathrm{4}}{\:\sqrt{\mathrm{5}}−\mathrm{1}}\:−\:\frac{\mathrm{4}}{\:\sqrt{\mathrm{5}}+\mathrm{1}} \\ $$$$=\:\mathrm{4}\left(\frac{\sqrt{\mathrm{5}}+\mathrm{1}−\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}\right)\:=\:\mathrm{2}…
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Question Number 166168 by alcohol last updated on 14/Feb/22 $${prove} \\ $$$$\underset{{r}=−\infty} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{x}\:+\:\left({r}+\frac{\mathrm{1}}{\mathrm{2}}\right)\pi}\:=\:{tan}\left({x}\right) \\ $$$$\left(\underset{{r}=−\infty} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{x}\:+\:{r}}\right)\left(\underset{{r}=−\infty} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{x}\:+\:{r}}\right)\:=\:−\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:\:\:\left(\:{r}\:=\:{odd}\right)\:\:\:\:\:\:\:\:\left({r}\:=\:{even}\right) \\ $$…
Question Number 166169 by cortano1 last updated on 14/Feb/22 $$\:\:\mathrm{sin}^{\mathrm{7}} \left(\mathrm{x}\right)+\frac{\mathrm{1}}{\mathrm{sin}\:^{\mathrm{3}} \left(\mathrm{x}\right)}=\mathrm{cos}\:^{\mathrm{7}} \left(\mathrm{x}\right)+\frac{\mathrm{1}}{\mathrm{cos}\:^{\mathrm{3}} \left(\mathrm{x}\right)} \\ $$$$ \\ $$ Answered by greogoury55 last updated on 14/Feb/22…