Question

If $f\left(x\right)={\cos }^{2}x+{\cos }^2(\frac{\pi }{3}+x)-\cos x\cdot \cos (\frac{\pi }{3}+x)$ then

• A. $f\left(x\right)$ is an even function
• B. $f\left(\frac{\pi }{8}\right)=f\left(\frac{\pi }{4}\right)$
• C. $f\left(x\right)$ is a constant function
• D. $f\left(x\right)$ is not a constant function

Answer 1

Answer: (A), (B), (C)

$f\left(x\right)$ is an even function
$f\left(x\right)$ is a constant function
$f\left(\frac{\pi }{8}\right)=f\left(\frac{\pi }{4}\right)$

${\cos }^2\left(x\right)+{[\frac{\cos \left(x\right)}{2}-\frac{\sqrt{3}\sin x}{2}]}^2-\cos x(\frac{\cos \left(x\right)}{2}-\frac{\sqrt{3}\sin x}{2})$

$={\cos }^2\left(x\right)+(\frac{\cos \left(x\right)}{2}-\frac{\sqrt{3}\sin x}{2})(\frac{\cos \left(x\right)}{2}-\frac{\sqrt{3}\sin x}{2}-\cos x)$

$={\cos }^2\left(x\right)+(\frac{\cos \left(x\right)}{2}-\frac{\sqrt{3}\sin x}{2})(-\frac{\cos \left(x\right)}{2}-\frac{\sqrt{3}\sin x}{2})$

$={\cos }^2\left(x\right)-(\frac{\cos \left(x\right)}{2}-\frac{\sqrt{3}\sin x}{2})(\frac{\cos \left(x\right)}{2}+\frac{\sqrt{3}\sin x}{2})$

$={\cos }^2\left(x\right)-\frac{{\cos }^2\left(x\right)}{4}+\frac{3{\sin }^2\left(x\right)}{4}$

$=\frac{3{\cos }^2\left(x\right)}{4}+\frac{3{\sin }^2\left(x\right)}{4}$

$=\frac{3}{4}$

Hence, $f\left(x\right)$ is a constant function.

Hence, options A, B and C are correct