Question

Solve the equation $si{n}^{4}x+co{s}^{4}x-2si{n}^{2}x+si{n}^{2}2x=0$

• A. $x=n\pi \pm \frac{1}{2}co{s}^{-1}(1-\sqrt{10}),nϵz$
• B. $x=n\pi \pm \frac{1}{2}si{n}^{-1}(1-\sqrt{10}),nϵz$
• C. $x=n\pi /2\pm co{s}^{-1}(1-\sqrt{10}),nϵz$
• D. $x=n\pi /2\pm si{n}^{-1}(1-\sqrt{10}),nϵz$

Answer 1

The correct option is A $x=n\pi \pm \frac{1}{2}co{s}^{-1}(1-\sqrt{10}),nϵz$

${\sin }^{4}x+{\cos }^{4}x-2{\sin }^{2}x+{\sin }^{2}2x=0$

$\Rightarrow 1+2{\sin }^{2}x{\cos }^{2}x-2{\sin }^{2}x+{\sin }^{2}2x=0$

$\Rightarrow 1+\frac{1}{2}{\sin }^{2}2x-2{\sin }^{2}x+{\sin }^{2}2x=0$

$\Rightarrow 1+\frac{3}{2}{\sin }^{2}2x-2{\sin }^{2}x=0\Rightarrow 1-2{\sin }^{2}x+\frac{3}{2}{\sin }^{2}2x=0$

$\Rightarrow \cos 2x+\frac{3}{2}{\sin }^{2}2x=0\Rightarrow \cos 2x+\frac{3}{2}(1-{\cos }^{2}2x)=0$

$\Rightarrow \cos 2x+\frac{3}{2}-\frac{3}{2}{\cos }^{2}2x=0\Rightarrow 3{\cos }^{2}2x-2\cos 2x-3=0$

$\Rightarrow \cos 2x=\frac{2\pm \sqrt{4+36}}{2}=\frac{2\pm 2\sqrt{10}}{2}=1\pm \sqrt{10}$

$\Rightarrow 2x-2n\pi \pm {\cos }^{-1}(1\pm \sqrt{10})\Rightarrow x=n\pi \pm \frac{1}{2}{\cos }^{-1}(1\pm \sqrt{10})$