Question

Fundamental period of $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$ is

• A. $2\pi$
• B. $\frac{3\pi }{2}$
• C. $\frac{\pi }{2}$
• D. $\pi$

Answer 1

The correct option is C $\frac{\pi }{2}$

Let $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$
$=({\sin }^{2}x+{\cos }^{2}x{)}^2-2{\sin }^{2}x{\cos }^{2}x=1-\frac{4{\sin }^{2}x{\cos }^{2}x}{2}=1-\frac{{\sin }^{2}2x}{2}=1-\frac{1}{4}(2{\sin }^{2}2x)=1-\left(\frac{1-\cos 4x}{4}\right)=\frac{3}{4}+\frac{1}{4}\cos 4x$
$\because$ fundamental period of $\cos x=2\pi$
Hence, fundamental period of function $=\frac{2\pi }{4}=\frac{\pi }{2}$