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  4. If cos 3 θ+ cos 3((2 π/3)+θ)+ cos 3((4 π/3)+θ) =a cos 3 θ, then a is equal to

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Q. If $\cos ^{3} \theta+\cos ^{3}\left(\frac{2 \pi}{3}+\theta\right)+\cos ^{3}\left(\frac{4 \pi}{3}+\theta\right) =a \cos 3 \theta$, then $a$ is equal to

TS EAMCET 2016

Solution:

We have,
$\cos ^{3} \theta+\cos ^{3}\left(\frac{2 \pi}{3}+\theta\right)+\cos ^{3}\left(\frac{4 \pi}{3}+\theta\right)=a \cos 3 \theta$
$\Rightarrow \frac{\cos 3 \theta+3 \cos \theta}{4}+\frac{\cos 3\left(\frac{2 \pi}{3}+\theta\right)+3 \cos \left(\frac{2 \pi}{3}+\theta\right)}{4}$
$+\frac{\cos 3\left(\frac{4 \pi}{3}+\theta\right)+3 \cos \left(\frac{4 \pi}{3}+\theta\right)}{4}=a \cos 3 \theta$
$\Rightarrow \cos 3 \theta+3 \cos \theta+\cos (2 \pi+3 \theta)+3 \cos \left(\frac{2 \pi}{3}+\theta\right)$
$+\cos (4 \pi+3 \theta)+3 \cos \left(\frac{4 \pi}{3}+\theta\right)=4 a \cos 3 \theta$
$\Rightarrow \cos 3 \theta+3 \cos \theta+\cos 3 \theta+\cos 3 \theta+3\left[\cos \left(\frac{2 \pi}{3}+\theta\right)\right.$
$\left.+\cos \left(\frac{4 \pi}{3}+\theta\right)\right]=4 a \cos 3 \theta$
$\Rightarrow 3 \cos 3 \theta+3 \cos \theta+3\left[2 \cos (\pi+\theta) \cos \frac{\pi}{3}\right]$
$=4 a \cos 3 \theta$
$\Rightarrow 3 \cos 3 \theta+3 \cos \theta+3 \times 2 \times(-\cos \theta) \times \frac{1}{2}$
$=4 a \cos 3 \theta$
$\Rightarrow 3 \cos 3 \theta+3 \cos \theta-3 \cos \theta=4 a \cos 3 \theta$
$\Rightarrow 3 \cos 3 \theta=4 a \cos 3 \theta $
$\Rightarrow a=\frac{3}{4}$