Difference of doing a PSD estimate of data and logarithmic transformed data?
The exercise is instructive. Since this is an exercise, I will not do the full problem.  I leave that to you.

grabbing the lynx data from

https://www.encyclopediaofmath.org/index.php/Canadian_lynx_data.

The data is counts and there are no years with a zero count.



and putting it into matlab and taking the log and subtracting the mean of the log,



Would you say that the data looks sine like ?

This technique works on certain kinds of data.  Think of it as a tool and is covered in a number of basic stats book.