Question

Find all solutions of $4co{s}^{2}xsinx-2si{n}^{2}x=3sinx$

• A. $x=2n\pi$ or$x=m\pi \pm (-1{)}^{m}si{n}^{-1}\left(\frac{-1\pm \sqrt{5}}{4}\right)$
• B. $x=n\pi$ or$x=2m\pi \pm (-1{)}^{m}si{n}^{-1}\left(\frac{-1\pm \sqrt{5}}{4}\right)$
• C. $x=n\pi$ or$x=m\pi \pm (-1{)}^{m}si{n}^{-1}\left(\frac{-1\pm \sqrt{5}}{4}\right)$
• D. none of these

Answer 1

Answer: (C)

$x=n\pi$ or$x=m\pi \pm (-1{)}^{m}si{n}^{-1}\left(\frac{-1\pm \sqrt{5}}{4}\right)$

$4{\cos }^{2}x\sin x-2{\sin }^{2}x=3\sin x\Rightarrow 4(1-{\sin }^{2}x)\sin x-2{\sin }^{2}x=3\sin x\Rightarrow 4\sin x-4{\sin }^{3}x-2{\sin }^{2}x=3\sin x\Rightarrow -4{\sin }^{3}x-2{\sin }^{2}x+\sin x=0\Rightarrow 4{\sin }^{3}x+2{\sin }^{2}x-\sin x=0\Rightarrow \sin x\left(4{\sin }^{2}x+2\sin x-1=0\right)$
$\Rightarrow \sin x=0$ or $4{\sin }^{2}x+2\sin x-1=0$
$\Rightarrow \sin x=0$ or $\sin x=\frac{-2\pm \sqrt{4+16}}{2\cdot 4}$
$\Rightarrow \sin x=0$ or $\sin x=\frac{-1\pm \sqrt{5}}{4}$
$\Rightarrow x=n\pi$ or $x=m\pi +{(-1)}^m{\sin }^{-1}\left(\frac{-1\pm \sqrt{5}}{4}\right)$