Question

Solution of the equation $ 4 \sin^4\,x+ \cos^4\,x = 1$ is

• A. $x=n\pi$
• B. $x=2n\pi\pm \cos^{-1}\left(\sqrt{\frac{3}{5}}\right)$
• C. $x=(2n+1)\frac{\pi}{2}$
• D. none of these.

Answer 1

Answer: (B)

$4 \, \sin^4 x + \cos^4 x = 1$
$\Rightarrow $ $4 (1 - \cos^2 x)^2 + \cos^4 x = 1$
$\Rightarrow $ $4(1 + \cos^4x - 2 \, \cos^2 x) + cos^4 x - 1 = 0 $
$\Rightarrow $ $5 \, \cos^4 x - 8 \, \cos^2 x + 3 = 0$
$\Rightarrow $ $(5 \, \cos^2 x - 3)(\cos^2 x - 1) = 0$
If $\cos^2x$ = 1, then $x= n\pi , n \in I$
If $\cos^2x = \frac{3}{2}$ , then $\cos x = \pm \sqrt{\frac{3}{5}} $
$\therefore $ $x =2 n \pi \pm cos^{-1} \sqrt{\frac{3}{5}}$
Hence $x = 2n\pi \pm \, \cos^{-1} \, \sqrt{\frac{3}{5}}$ gives the general value.