Question

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

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

Answer 1

Answer: (B)

$\frac{\pi }{2}$

${\sin }^{4}x+{\cos }^{4}x={\sin }^{4}x+[1-{\sin }^{2}x{]}^2$

$={\sin }^{4}x+1-2{\sin }^{2}x+{\sin }^{4}x$

$=2{\sin }^{4}x-2{\sin }^{2}x+1$

$=\frac{1}{2}\times [1-\cos (2x){]}^2-1+\cos (2x)+1$

$=\frac{1-2\cos \left(2x\right)+{\cos }^2\left(2x\right)+2\cos \left(2x\right)}{2}$

$=\frac{1}{2}+\frac{1+cos\left(4x\right)}{4}$

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

Since the period of $\cos \left(x\right)$ is $2\pi$, the period of $\cos \left(4x\right)$ becomes $\frac{2\pi }{4}=\frac{\pi }{2}$