Question

If $\cos x\ne \frac{1}{2},$ then the solutions of $\cos x+\cos 2x+\cos 3x=0$ are

• A. $2n\pi \pm \frac{\pi }{4},n\in Z$
• B. $2n\pi \pm \frac{\pi }{3},n\in Z$
• C. $2n\pi \pm \frac{\pi }{6},n\in Z$
• D. $2n\pi \pm \frac{\pi }{2},n\in Z$

Answer 1

Answer: (A)

$\cos x+\cos 3x+cos2x=0$
$\Rightarrow$ $2\cos 2x\cos x+\cos 2x=0$
$\Rightarrow$ $\cos 2x(2cosx+1)=0$
$\Rightarrow$ $\cos 2x=0\left(\because \cos x\ne -\frac{1}{2}\right)$
$\Rightarrow$ $2x=\frac{\pi }{2}$
$\Rightarrow$ $x=\frac{\pi }{4}$
$\therefore$ General value is $2n\pi \pm \frac{\pi }{4},n\in Z$