Question

The solution of the equation $\cos 3\theta =4\cos \theta \cos (\theta +x)\cos (\theta -x)(0<\theta <\pi )$ is

• A. $k\pi +\frac{\pi }{3}$
• B. $2k\pi$
• C. $k\pi -\frac{\pi }{3}$
• D. None of these

Answer 1

Answer: (A), (C)

The correct options are
A $k\pi +\frac{\pi }{3}$
C $k\pi -\frac{\pi }{3}$

$\cos 3\theta =4\cos \theta cos(\theta +x)\cos (\theta -x)$

$\Rightarrow 4{\cos }^3\theta -3\cos \theta =4\cos \theta \cos (\theta +x)\cos (\theta -x)$

$\Rightarrow 4{\cos }^2\theta -3=4({\cos }^2\theta -{\sin }^{2}x)$

$\Rightarrow {\sin }^{2}x=\frac{3}{4}={\left(\frac{\sqrt{3}}{2}\right)}^2={sin}^2\frac{\pi }{3}$

$\Rightarrow x=k\pi \pm \frac{\pi }{3}$