Question

The period of the function $f(x)={\sin }^{4}x+{\cos }^{4}x$ is:

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $2\pi$
• D. none of these

Answer 1

Answer: (B)

The period ${\sin }^{4}x$
or ${\cos }^{4}x$ is $\frac{\pi }{2}$ .
$f(x)={\sin }^{4}x+{\cos }^{4}x$
$={({\sin }^{2}x+{\cos }^{2}x)}^2-2{\sin }^{2}x{\cos }^{2}x$
$=1-\frac{1}{2}{(\sin 2x)}^2=1-\frac{1}{2}\left[\frac{1-\cos 4x}{2}\right]$
$=\frac{3}{4}+\frac{1}{4}\cos 4x$
Since, $\cos x$ is periodic with period $2\pi$ .
$\therefore$ The period of $f(x)=\frac{2\pi }{4}=\frac{\pi }{2}$ .
Alternative Solution: Since the period of ${\sin }^{4}x$ and ${\cos }^{4}x$ is $\frac{\pi }{2}$ .
$\therefore$ Period of $f(x)={\sin }^{4}x+{\cos }^{4}x$ is the $LCM<br/>$ of the period of ${\sin }^{4}x$ and
${\cos }^{4}x$ .
$\therefore$ Required period is $\frac{\pi }{2}$ .