Question Number 159049 by mnjuly1970 last updated on 12/Nov/21
$$ \\ $$$$\:\:\:\:\:\:{solve}\:{the}\:{following}\:{equation}: \\ $$$$\: \\ $$$${sin}^{\:\mathrm{6}} \left({x}\right)\:+{cos}^{\:\mathrm{6}} \left({x}\right)+{sin}^{\mathrm{4}} \left({x}\right)+{cos}^{\:\mathrm{4}} \left({x}\right)=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$ \\ $$
Answered by gsk2684 last updated on 12/Nov/21
$$\Rightarrow\left(\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{cos}\:^{\mathrm{2}} {x}\right)^{\mathrm{3}} −\mathrm{3}\:\mathrm{sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x}\left(\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{cos}\:^{\mathrm{2}} {x}\right) \\ $$$$+\left(\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{cos}\:^{\mathrm{2}} {x}\right)^{\mathrm{2}} −\mathrm{2}\:\mathrm{sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\Rightarrow\mathrm{1}−\mathrm{3sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x}+\mathrm{1}−\mathrm{2sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\Rightarrow\frac{\mathrm{5}}{\mathrm{4}}=\mathrm{5}\:\mathrm{sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x} \\ $$$$\Rightarrow\mathrm{1}=\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{x} \\ $$$$\therefore\:\mathrm{2}{x}=\left(\mathrm{2}{n}+\mathrm{1}\right)\frac{\pi}{\mathrm{2}},{n}\in{Z} \\ $$$$\:\:{x}=\left(\mathrm{2}{n}+\mathrm{1}\right)\frac{\pi}{\mathrm{4}},{n}\in{Z} \\ $$$$\:\:\:\:\:\:…..{gsk}…. \\ $$
Commented by mnjuly1970 last updated on 12/Nov/21
$${thanks}\:{alot}\:{sir}\:{g}.{s}.{k} \\ $$
Commented by gsk2684 last updated on 13/Nov/21
$${welcome} \\ $$
Answered by MJS_new last updated on 12/Nov/21
$$\mathrm{sin}\:{x}\:={s}\:\wedge\:\mathrm{cos}\:{x}\:=\sqrt{\mathrm{1}−{s}^{\mathrm{2}} } \\ $$$$\Rightarrow \\ $$$${s}^{\mathrm{4}} −{s}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{4}}=\mathrm{0} \\ $$$$\left({s}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow \\ $$$$\mathrm{sin}\:{x}\:=\pm\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\Rightarrow \\ $$$${x}=−\frac{\pi}{\mathrm{4}}+\frac{{n}\pi}{\mathrm{2}} \\ $$