Question

If ${(\cos \theta +\cos 2\theta )}^3={\cos }^3\theta +{\cos }^{3}2\theta$, then the least positive value of $\theta$ is equal to

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (B)

$\frac{\pi }{4}$

Upon simplifying, we get
$co{s}^3\theta +co{s}^{3}2\theta +3(cos\theta .cos2\theta )(cos\theta +cos2\theta )=co{s}^3\theta +co{s}^{3}2\theta$.
Hence
$3(cos\theta .cos2\theta )(cos\theta +cos2\theta )=0$
Or
$6cos\theta .cos2\theta .cos\left(\frac{3\theta }{2}\right).cos\left(\frac{\theta }{2}\right)=0$
Hence
$\theta =(2k+1)\frac{\pi }{2}$.
$\theta =(2m+1)\frac{\pi }{4}$.
$\theta =(2n+1)\frac{\pi }{3}$
And
$\theta =(2t+1)\pi$.
Hence least possible value is $\frac{\pi }{4}$.