Question

Solve $\frac{\sqrt{3}}{2}\sin x-\cos x=co{s}^{2}x$.

Answer 1

$\frac{1}{2}\sqrt{3}\sin x=\cos x+{\cos }^{2}x$. Square
$3(1-{\cos }^{2}x)=4({\cos }^{2}x+2{\cos }^{3}x+{\cos }^{4}x)$
or $4{\cos }^{4}x+8co{s}^3+7{\cos }^{2}x-3=0$
Clearly $\cos x=-1$ satisfies above, hence it can be written as
$(\cos x+1)(4{\cos }^{3}x+4{\cos }^{2}x+3\cos x-3)=0$
Again $\cos x=\frac{1}{2}$ satisfies the $2$nd factor
$(\cos x+1)(2\cos x-1)(2{\cos }^{2}x+3\cos x+3)=0$
$\therefore \cos x=-1=\cos \pi$
$\therefore x=2n\pi +\pi =(2n+1)\pi$
$\cos x=1/2=\cos \left(\pi /3\right)$
$\therefore x=2n\pi \pm \pi /3$
$2{\cos }^{2}x+3\cos x+3=0$,
$B^2-4AC=9-\mathrm{4.2.3}=-$ive.
Hence this factor does not give any real values of $\cos x$.