Question

If $\sin ^{2} x-\cos x=1 / 4$, if the sum of values of $x$ between $0$ and $2 \pi$ is $k \pi$ then find $k$.

Answer 1

$\sin ^{2} x-\cos x=\frac{1}{4}$
$\Rightarrow 1-\cos ^{2} x-\cos x=\frac{1}{4}$
$\Rightarrow 4 \cos ^{+2} x+ y \cos x-3=0$
$\Rightarrow(2 \cos x+3)(2 \cos x-1)=0$
$\Rightarrow 2 \cos x=1(\because \cos x \neq-3 / 2)$
$\Rightarrow \cos x =1 / 2$
$\Rightarrow \cos x=\cos \pi / 3$
$\Rightarrow x=2 n\, \pi \frac{\pm \pi}{3}, n \in I$
$\therefore x =\frac{\pi}{3}, \frac{5 \pi}{3} \in[0,2 \pi]$