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
$ of the period of $ {{\sin }^{4}}x $ and
$ {{\cos }^{4}}x $ .
$ \therefore $ Required period is $ \frac{\pi }{2} $ .