Question

Assertion :$f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$ 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

$\because$ $f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$
$=\frac{1}{2}\{2{\sin }^{2}x+2{\sin }^2(x+\frac{\pi }{3})+2\cos x\cos (x+\frac{\pi }{3})\}$
$=\frac{1}{2}\{1-\cos 2x+1-\cos (2x+\frac{2\pi }{3})+\cos (2x+\frac{\pi }{3})+\cos \frac{\pi }{3}\}$
$=\frac{1}{2}\{\frac{5}{2}-\cos 2x-\cos (2x+\frac{2\pi }{3})+\cos (2x+\frac{\pi }{3})\}$
$=\frac{1}{2}\{\frac{5}{2}-(2\cos (x+\frac{\pi }{3}).\cos \frac{\pi }{3})+\cos (2x+\frac{\pi }{3})\}$
$=\frac{1}{2}\left(\frac{5}{2}\right)=\frac{5}{4}$
$\therefore$ $f^{\prime }\left(x\right)=0$