Question

The period of the function
$f\left(x\right)=c^{\left[{\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})\right]}$
is (where $c$ is constant)

• A. $1$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. Cannot be determined

Answer 1

The correct option is D Cannot be determined

The above expression can be simplified by expansion.Therefore

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

$=c^{{\sin }^{2}x+\frac{{\sin }^{2}x}{4}+\frac{3{\cos }^{2}x}{4}+\frac{\sqrt{3}\sin x\cos x}{2}+\frac{{\cos }^2}{2}-\frac{\sqrt{3}\sin x\cos x}{2}}$

$=c^{{\sin }^{2}x+\frac{{\sin }^{2}x}{4}+\frac{3{\cos }^{2}x}{4}+\frac{{\cos }^{2}x}{2}}$

$=c^{\left(5{\sin }^{2}x+5{\cos }^{2}x\right)/4}$

$=c^{5/4}$

Which is a constant function, therefore its period can not be determined.

Hence, option 'D' is correct.