Question

${\cos }^3\theta +{\cos }^3({120}^{\circ }+\theta )+co{s}^3({120}^{\circ }-\theta )=$

• A. $\frac{3}{4}\sin 3\theta$
• B. $\frac{3}{4}\cos 3\theta$
• C. $\frac{3}{4}\tan 3\theta$
• D. $\frac{3}{4}\cot 3\theta$

Answer 1

The correct option is B $\frac{3}{4}\cos 3\theta$

We know that $\cos 3\theta =4{\cos }^3\theta -3\cos \theta$

or $4{\cos }^3\theta =\frac{1}{4}\cos 3\theta +\frac{3}{4}\cos \theta$

${\cos }^3\theta +{\cos }^3({120}^{\circ }+\theta )+{\cos }^3({120}^{\circ }-\theta )$

$={\cos }^3\theta +{\cos }^3({180}^{\circ }-({60}^{\circ }-\theta ))+{\cos }^3({180}^{\circ }-({60}^{\circ }+\theta ))$

$={\cos }^3\theta -{\cos }^3({60}^{\circ }-\theta )-{\cos }^3({60}^{\circ }+\theta )$

$=\frac{1}{4}\cos 3\theta +\frac{3}{4}\cos \theta -\frac{1}{4}\cos 3({60}^{\circ }-\theta )-\frac{3}{4}\cos ({60}^{\circ }-\theta )-\frac{1}{4}\cos 3({60}^{\circ }+\theta )-\frac{3}{4}\cos ({60}^{\circ }+\theta )$

$=\frac{1}{4}\cos 3\theta +\frac{3}{4}\cos \theta -\frac{1}{4}\cos ({180}^{\circ }-3\theta )-\frac{3}{4}\cos ({60}^{\circ }-\theta )-\frac{1}{4}\cos ({180}^{\circ }+3\theta )-\frac{3}{4}\cos ({60}^{\circ }+\theta )$

$=\frac{1}{4}\cos 3\theta +\frac{3}{4}\cos \theta +\frac{1}{4}\cos 3\theta -\frac{3}{4}\cos ({60}^{\circ }-\theta )+\frac{1}{4}\cos 3\theta -\frac{3}{4}\cos ({60}^{\circ }+\theta )$

$=\frac{3}{4}\cos 3\theta +\frac{3}{4}\cos \theta -\frac{3}{4}\cos ({60}^{\circ }-\theta )-\frac{3}{4}\cos ({60}^{\circ }+\theta )$

$=\frac{3}{4}\cos 3\theta +\frac{3}{4}\cos \theta -\frac{3}{4}[\cos ({60}^{\circ }-\theta )+\cos ({60}^{\circ }+\theta )]$

$=\frac{3}{4}\cos 3\theta +\frac{3}{4}\cos \theta -\frac{3}{4}\left[2\cos \left(\frac{60-\theta +60+\theta }{2}\right)\cos \left(\frac{60-\theta -60-\theta }{2}\right)\right]$

$=\frac{3}{4}\cos 3\theta +\frac{3}{4}\cos \theta -\frac{3}{4}\left[2\cos {60}^{\circ }\cos \theta \right]$

$=\frac{3}{4}\cos 3\theta +\frac{3}{4}\cos \theta -\frac{3}{4}[2\times \frac{1}{2}\cos \theta ]$

$=\frac{3}{4}\cos 3\theta +\frac{3}{4}\cos \theta -\frac{3}{4}\cos \theta$

$=\frac{3}{4}\cos 3\theta$