Answer
The value of given expression is $1$.
Work Step by Step
The sine function has periodicity of $2\pi $.
So,
$\begin{align}
& f\left( a+2\pi \right)=f\left( a \right) \\
& f\left( a+4\pi \right)=f\left( a \right) \\
& f\left( a+6\pi \right)=f\left( a \right) \\
\end{align}$
Put all the values of the above in the main expression:
$\begin{align}
& f\left( a \right)+f\left( a+2\pi \right)+f\left( a+4\pi \right)+f\left( a+6\pi \right)=f\left( a \right)+f\left( a \right)+f\left( a \right)+f\left( a \right) \\
& =4f\left( a \right) \\
& =4\times \frac{1}{4} \\
& =1
\end{align}$
Thus, the value of the given expression is $1$.
