Question Number 89079 by john santu last updated on 15/Apr/20
$$\left(\mathrm{cos}\:\mathrm{4}{x}+\mathrm{1}\right)\left(\mathrm{cos}\:\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{cos}\:{x}+\mathrm{1}\right)=\:\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\mathrm{0}\:\leqslant\:{x}\:\leqslant\:\mathrm{2}\pi \\ $$
Answered by TANMAY PANACEA. last updated on 15/Apr/20
$$\mathrm{2}{cos}^{\mathrm{2}} \mathrm{2}{x}.\mathrm{2}{cos}^{\mathrm{2}} {x}.\mathrm{2}{cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\mathrm{8}{p}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{8}} \\ $$$${p}={cos}\frac{{x}}{\mathrm{2}}.{cosx}.{cos}\mathrm{2}{x} \\ $$$$\mathrm{2}{sin}\frac{{x}}{\mathrm{2}}.{p}={sinxcosxcos}\mathrm{2}{x} \\ $$$$\mathrm{2}^{\mathrm{2}} {sin}\frac{{x}}{\mathrm{2}}.{p}={sin}\mathrm{2}{x}.{cos}\mathrm{2}{x} \\ $$$$\mathrm{2}^{\mathrm{3}} {sin}\frac{{x}}{\mathrm{2}}.{p}={sin}\mathrm{4}{x} \\ $$$${p}=\frac{{sin}\mathrm{4}{x}}{\mathrm{8}{sin}\frac{{x}}{\mathrm{2}}} \\ $$$${so}\:\mathrm{8}{p}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\mathrm{8}\left(\frac{{sin}\mathrm{4}{x}}{\mathrm{8}{sin}\frac{{x}}{\mathrm{2}}}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{8}} \\ $$$${sin}^{\mathrm{2}} \mathrm{4}{x}={sin}^{\mathrm{2}} \frac{{x}}{\mathrm{2}} \\ $$$$\mathrm{2}{sin}^{\mathrm{2}} \mathrm{4}{x}=\mathrm{2}{sin}^{\mathrm{2}} \frac{{x}}{\mathrm{2}} \\ $$$$\mathrm{1}−\mathrm{2}{sin}^{\mathrm{2}} \mathrm{4}{x}=\mathrm{1}−\mathrm{2}{sin}^{\mathrm{2}} \frac{{x}}{\mathrm{2}} \\ $$$${cos}\mathrm{8}{x}={cosx} \\ $$$$\mathrm{8}{x}=\mathrm{2}{n}\pi\pm{x} \\ $$$$\mathrm{7}{x}=\mathrm{2}{n}\pi\:\:\:\:{or}\:\:\:\:\mathrm{9}{x}=\mathrm{2}{n}\pi \\ $$$${x}=\frac{\mathrm{2}\pi}{\mathrm{7}}\:\:\:\:\:\:\:{x}=\frac{\mathrm{2}\pi}{\mathrm{9}} \\ $$$${x}=\frac{\mathrm{4}\pi}{\mathrm{7}}\:\:\:\:\:\:{x}=\frac{\mathrm{4}\pi}{\mathrm{9}} \\ $$$${x}=\frac{\mathrm{6}\pi}{\mathrm{7}}\:\:\:\:{x}=\frac{\mathrm{6}\pi}{\mathrm{9}} \\ $$$${x}=\frac{\mathrm{8}\pi}{\mathrm{7}}\:\:\:\:\:{x}=\frac{\mathrm{8}\pi}{\mathrm{9}} \\ $$$${x}=\frac{\mathrm{10}\pi}{\mathrm{7}}\:\:\:\:{x}=\frac{\mathrm{10}\pi}{\mathrm{9}} \\ $$$${x}=\frac{\mathrm{14}\pi}{\mathrm{7}}\:\:\:\:\:\:{x}=\frac{\mathrm{14}\pi}{\mathrm{9}} \\ $$$$\:\:\:\:\:\:\:\:−\:\:\:\:\:\:\:{x}=\frac{\mathrm{16}\pi}{\mathrm{9}} \\ $$$${x}=−\:\:\:\:\:\:\:{x}=\frac{\mathrm{18}\pi}{\mathrm{9}} \\ $$$${pls}\:{check} \\ $$
Commented by john santu last updated on 15/Apr/20
$${yes}\:{sir}.\:{our}\:{correct}\:{is}\:{same} \\ $$
Answered by john santu last updated on 15/Apr/20