Question

Assertion :${\cos }^3\alpha +{\cos }^3(\alpha +\frac{2\pi }{3})+{\cos }^3(\alpha +\frac{4\pi }{3})=3\cos \alpha \cos (\alpha +\frac{2\pi }{3})\cos (\alpha +\frac{4\pi }{3})$ Reason: If $a+b+c=0\iff a^3+b^3+c^3=3abc$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ ... $\left(i\right)$

If $(a+b+c)=0$ then $a^3+b^3+c^3-3abc=0$

$⟹a^3+b^3+c^3=3abc$

Thus, the reason is correct.
Assertion

$\cos \alpha +\cos (\alpha +\frac{2\pi }{3})+\cos (\alpha +\frac{4\pi }{3})$

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

$=\cos \alpha +2\cos (\alpha +\pi )\cos \left(\frac{\pi }{3}\right)$ ..... [Using $\cos a+\cos b$ formula]

$=\cos \alpha -2\cos \alpha \cdot \frac{1}{2}$

$=\cos \alpha -\cos \alpha =0$

Using the reason, we get

${\cos }^3\alpha +{\cos }^3(\alpha +\frac{2\pi }{3})+{\cos }^3(\alpha +\frac{4\pi }{3})=3\cos \alpha \cos (\alpha +\frac{2\pi }{3})\cos (\alpha +\frac{4\pi }{3})$

Hence, both reason and assertion are correct and Reason is the correct explanation of Assertion.