Question

Assertion :Let $f\left(x\right)={\cos }^{2}x+{\cos }^2(x+\frac{\pi }{3})+{\cos }^2(x-\frac{\pi }{3})$, then $f^{\prime }\left(x\right)=0$. Reason: Derivative of a constant function is always zero.

• 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

$f\left(x\right)={\cos }^{2}x+{\cos }^2(\frac{\pi }{3}+x)+{\cos }^2(x-\frac{\pi }{3})$
$={\cos }^{2}x+{\cos }^2(x+\frac{\pi }{3})+1-{\sin }^2(x-\frac{\pi }{3})$
$=1+{\cos }^{2}x+{\cos }^2(x+\frac{\pi }{3})-{\sin }^2(x-\frac{\pi }{3})$
$=1+{\cos }^{2}x+\cos {120}^{\circ }\cos 2x$
$=1+{\cos }^{2}x-\frac{1}{2}\cos 2x$
$=1+{\cos }^{2}x-\frac{1}{2}(2{\cos }^{2}x-1)$
$\Rightarrow$ $f\left(x\right)=\frac{3}{2}$, which is a constant
$\therefore$ $f^{\prime }\left(x\right)=\frac{d}{dx}\left(\frac{3}{2}\right)=0$