Question

Solve the following equation:
${\sin }^{4}x+{\cos }^{4}x=\sin x\cos x.$

Answer 1

${({\sin }^{2}x+{\cos }^{2}x)}^2-{2\sin }^{2}x{\cos }^{2}x=\sin x\cos x$

Let $\sin x\cos x=z.$

$1-{2z}^2=z$

${2z}^2+z-1=0$

${2z}^2+2z-z-1=0$

$(z+1)(2z-1)=0$

So, $z=-1,\sin x\cos x=-1,\sin 2x=-2.\text{[Thus no value possible.]}$

$2z-1=0,\sin 2x=1,2x=2n\pi +\frac{\pi }{2},x=n\pi +\frac{\pi }{4}.$