OK, I've been working on this... I've got a few approaches. The first is this:

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From theory, we know that the fourier transform of a rectangle function is a sinc:

So, if the fourier transform of is , using the symmetry property (duality):

We can get

From the similarity property of the fourier transform, we get

so, scaling t in by a gives

so to find , substitute for a in the previous line

To find 2sinc(st) from the original question, multiply both sides by two using the linearity property.

So, if the fourier transform of is , using the symmetry property (duality):

We can get

From the similarity property of the fourier transform, we get

so, scaling t in by a gives

so to find , substitute for a in the previous line

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This works for me. Do you think this would suffice to answer the question posed above?

The other approach.. I'll split into a seperate post later on, got to get to work!