Question

If $4{\sin }^{4}x+{\cos }^{4}x=1$, then $x$ is equal to $\left(nϵz\right)$

• A. $n\pi$
• B. $n\pi \pm {\sin }^{-1}\sqrt{\frac{2}{5}}$
• C. $\frac{2n\pi }{3}$
• D. $2n\pi \pm \frac{\pi }{4}$

Answer 1

Answer: (A), (B)

$n\pi \pm {\sin }^{-1}\sqrt{\frac{2}{5}}$
$n\pi$

$4{\sin }^{4}x+{\cos }^{4}x=1=4{\sin }^{4}x+(1-{\sin }^{2}x{)}^2$
$\Rightarrow 4{\sin }^{4}x+1+{\sin }^{4}x-2{\sin }^{2}x=1$
$\Rightarrow 5{\sin }^{4}x-2{\sin }^{2}x+1=1$
$\Rightarrow {\sin }^{2}x(5{\sin }^{2}x-2)=0$
$\Rightarrow \sin x=0or\sin x=\sqrt{\frac{2}{5}}$
$\Rightarrow x=n\pi orx=n\pi \pm {\sin }^{-1}\sqrt{\frac{2}{5}}$