Question

Of the given functions which of the following is not periodic
$\begin{align}
  & a)\left| \sin 3x \right|+{{\sin }^{2}}x \\
 & b)\cos \sqrt{x}+{{\cos }^{2}}x \\
 & c)\cos 4x+{{\tan }^{2}}x \\
 & d)\cos 2x+\sin x \\
\end{align}$

Answer 1

Hint: Now we know that a function $f(x)+g(x)$ is periodic if $f(x)$ and $g(x)$ are both periodic. Also we know that trigonometric functions are periodic and also squares of trigonometric functions are periodic. With these results we will check each option.

Complete step-by-step answer:
Now first let us understand what periodic functions are. Periodic functions are nothing but the functions which repeat the same values after a time period T.
Hence if $f\left( x \right)$ is periodic then we have $f\left( x+T \right)=f\left( x \right)$ and T is called period of function
Now we know that all trigonometric functions are periodic.
Now first let us consider option $a)\left| \sin 3x \right|+{{\sin }^{2}}x$
Now let us check if the function $\left| \sin 3x \right|+{{\sin }^{2}}x$.
Now we know that a function $f(x)+g(x)$ is periodic if $f(x)$ and $g(x)$ are both periodic.
For this function to be periodic $\left| \sin 3x \right|$ and ${{\sin }^{2}}x$ should both be periodic.
Now we know that modulus of sine and cosine functions are periodic.
And ${{\sin }^{n}}x,{{\cos }^{n}}x,{{\tan }^{n}},{{\cot }^{n}},{{\sec }^{n}}x,\cos e{{c}^{n}}x$ are also periodic function.
Hence we have $\left| \sin 3x \right|$ and ${{\sin }^{2}}x$ both as periodic functions.
Hence $\left| \sin 3x \right|+{{\sin }^{2}}x$ is a periodic function.
Now first let us consider option $b)\cos \sqrt{x}+{{\cos }^{2}}x$
Now let us check if the function $\cos \sqrt{x}+{{\cos }^{2}}x$.
Now we know that a function $f(x)+g(x)$ is periodic if $f(x)$ and $g(x)$ are both periodic.
For this function to be periodic $\cos \sqrt{x}$ and ${{\cos }^{2}}x$ should both be periodic.
And ${{\sin }^{n}}x,{{\cos }^{n}}x,{{\tan }^{n}},{{\cot }^{n}},{{\sec }^{n}}x,\cos e{{c}^{n}}x$ are also periodic function.
Hence ${{\cos }^{2}}x$ is a periodic function.
Consider $\cos \sqrt{x}$ also to be periodic, then we know that
$\cos \left( \sqrt{x+T} \right)=\cos \sqrt{x}$
Now at x = 0 we get
$\begin{align}
  & \cos \sqrt{T}=\cos 0 \\
 & \Rightarrow \cos \sqrt{T}=1 \\
 & \Rightarrow \sqrt{T}=2{{n}_{1}}\pi ,{{n}_{1}}\in Z..............(1) \\
\end{align}$
And if we put x = T we get,
$\begin{align}
  & \cos \sqrt{T+T}=\cos \sqrt{T} \\
 & \cos \sqrt{2T}=\cos \sqrt{T} \\
\end{align}$
But we got the value of $\cos \sqrt{T}=1$
Hence using this we get
$\begin{align}
  & \cos \sqrt{2T}=1 \\
 & \Rightarrow \sqrt{2T}=2{{n}_{2}}\pi ,{{n}_{2}}\in Z.................(2) \\
\end{align}$
Hence dividing (2) from (1) we get
$\begin{align}
  & \dfrac{\sqrt{2T}}{\sqrt{T}}=\dfrac{2{{n}_{2}}\pi }{2{{n}_{1}}\pi } \\
 & \sqrt{2}=\dfrac{{{n}_{2}}}{{{n}_{1}}} \\
\end{align}$
But this is a contradiction since we have ${{n}_{1}},{{n}_{2}}$ as integers an \[\sqrt{2}\] is irrational and we know that irrational numbers cannot be represented in the form of $\dfrac{p}{q}$ where p and q are integers.
Hence $\cos \sqrt{x}$ is not a periodic function
Hence $\cos \sqrt{x}+{{\cos }^{2}}x$ is not a periodic function.
Now first let us consider option $c)\cos 4x+{{\tan }^{2}}x$
Now let us check if the function $\cos 4x+{{\tan }^{2}}x$.
Now we know that a function $f(x)+g(x)$ is periodic if $f(x)$ and $g(x)$ are both periodic.
For this function to be periodic $\cos 4x$ and ${{\tan }^{2}}x$ should both be periodic.
Now we know that the functions $\sin ax,\cos ax$ are periodic functions.
And ${{\sin }^{n}}x,{{\cos }^{n}}x,{{\tan }^{n}},{{\cot }^{n}},{{\sec }^{n}}x,\cos e{{c}^{n}}x$ are also periodic function.
Hence ${{\tan }^{2}}x$ is periodic function and $\cos 4x$ is also a periodic function.
Hence $\cos 4x+{{\tan }^{2}}x$ is a periodic function.
Now first let us consider option $d)\cos 2x+\sin x$
Now let us check if the function $\cos 2x+\sin x$.
Now we know that a function $f(x)+g(x)$ is periodic if $f(x)$ and $g(x)$ are both periodic.
For this function to be periodic $\cos 2x$ and $\sin x$ should both be periodic.
Now we know that trigonometric functions of the form $\cos ax$ and $\sin x$ are periodic
Hence we have $\cos 2x$ and $\sin x$ both as periodic functions.
Hence $\cos 2x+\sin x$ is a periodic function.

So, the correct answer is “Option b”.

Note: Here note that $\cos x$ is periodic and $\cos \sqrt{x}$ is non periodic. We have $\cos ax$ to be periodic and ${{\cos }^{n}}x$ to be periodic we don’t know if $\cos {{x}^{n}}$ is periodic hence we will have to check it separately.