Question

Solve: $4{\sin }^{4}x+{\cos }^{4}x=1$.

Answer 1

The given equation can be put in the form $4{\sin }^{4}x=1-{\cos }^{4}x=(1-{\cos }^{2}x)(1+{\cos }^{2}x)$

or ${\sin }^{2}x[4{\sin }^{2}x-1-(1-{\sin }^{2}x\left)\right]=0$

or ${\sin }^{2}x(5{\sin }^{2}x-2)=0$

$\therefore \sin x=0$ or $\sin x=\pm \sqrt{2/5}$

or $x=n\pi$ or $x=n\pi \pm \alpha$,

where $\sin \alpha =\sqrt{2/5}$.