Answer
The distance between $M$ and $N$ is $R~[1 - cos(\frac{\theta}{2})]$
Work Step by Step
We can find an expression for the length of the line $CN$:
$\frac{CN}{R} = cos(\frac{\theta}{2})$
$CN = (R)~cos(\frac{\theta}{2})$
Note that the length of the line $CM$ is equal to $R$. Note also that the distance between $M$ and $N$ is $CM-CN$. We can find an expression for the distance between $M$ and $N$:
$CM - CN = R - (R)~cos(\frac{\theta}{2})$
$CM - CN = R~[1 - cos(\frac{\theta}{2})]$
The distance between $M$ and $N$ is $R~[1 - cos(\frac{\theta}{2})]$
