I learned that a modified wave number, $k'$, is the number that satisfies the following condition
$$f'_j = ik'f_j$$
However, there is a problem while I estimate a modified number of the below function
$$f(x) = 5 \cos(x) + \cos(10x)$$
I used the first order forward difference. So, I estimated the modified number like
\begin{align} f'_j &\approx \frac{f_{j+1} - f_j}{h} \\ &= \frac{5(\cos(x+h) - \cos(x)) + \cos(10(x+h)) - \cos(10x)}{h} \\ &= \frac{-10 \sin \left(x + \frac{h}{2} \right) \sin \left(\frac{h}{2} \right) - 2 \sin(10x+5h) \sin(5h)}{h} \end{align}
Then, I don't know how to rearrange RHS in order to get the form $5 \cos(x) + \cos(10x)$. Is my approach correct? or, Do I have to use other approach?