Question

The period of ${{\sin }^{4}}x+{{\cos }^{4}}x$ is.
A. $\dfrac{\pi }{2}$
B. $\pi $
C. $2\pi $
D. None of these

Answer 1

Hint: We will be using the concept of function to solve the problem. We will be using the periodicity of sine and cosine function and will also be utilizing the general condition for any function to be periodic.

Complete step-by-step solution -

We have been given a function $f\left( x \right)={{\sin }^{4}}x+{{\cos }^{4}}x$.
Now, we have to find the period of $f\left( x \right)$.
We know that periodic functions are those who repeat their value after a fixed constant interval called period.
In generally a function $f\left( x \right)$ such that,
$f\left( T+x \right)=f\left( x \right)$
Then T is the period of the function. For example, if
$f\left( x \right)=\sin x$
We know that $\sin \left( 2\pi +x \right)=\sin x$
Also, if $f\left( x \right)=\sin x$
Then,
$\begin{align}
  & f\left( \pi +x \right)=\sin {{\left( \pi +x \right)}^{2}} \\
 & ={{\left( -\sin \left( x \right) \right)}^{2}} \\
 & ={{\sin }^{2}}x \\
 & \Rightarrow f\left( \pi +x \right)=f\left( x \right) \\
\end{align}$
And therefore the period of ${{\sin }^{2}}x\ is\ \pi $.
Similarly, the period of ${{\cos }^{2}}x$ is also $\pi $.
Now, we have to find the period of $f\left( x \right)={{\sin }^{4}}x+{{\cos }^{4}}x$.
$\begin{align}
  & f\left( x \right)={{\sin }^{4}}x+{{\cos }^{4}}x \\
 & f\left( T+x \right)={{\sin }^{4}}\left( T+x \right)+{{\cos }^{4}}\left( T+x \right) \\
\end{align}$
Now, if we put $T=\dfrac{\pi }{2}$, we see that,
$f\left( \dfrac{\pi }{2}+x \right)={{\sin }^{4}}\left( \dfrac{\pi }{2}+x \right)+{{\cos }^{4}}\left( \dfrac{\pi }{2}+x \right)$
Also, we know that,
$\begin{align}
  & \sin \left( \dfrac{\pi }{2}+x \right)=\sin \left( x \right) \\
 & \cos \left( \dfrac{\pi }{2}+x \right)=-\cos \left( x \right) \\
\end{align}$
$\Rightarrow f\left( \dfrac{\pi }{2}+x \right)={{\cos }^{4}}x+{{\sin }^{4}}x$
Now, since,
$f\left( x \right)=f\left( \dfrac{\pi }{2}+x \right)$
Therefore, the period is $\dfrac{\pi }{2}$.

Note: To solve these types of functions it is advised to remember the period of function like sin x, cos x which is $2\pi $. Also, it is noted that the for ${{\sin }^{4}}x+{{\cos }^{4}}xf\left( \pi +x \right)=f\left( x \right)$. Also, but $\pi $ is still not period because period is always the least value of T which satisfies $f\left( T+x \right)=f\left( x \right)$