I Trouble with Product of Green's Functions

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Hi all,
Consider the following Green's function:
1698539684976.png

where ##\Theta(t)## is the Heaviside step function and ##\tilde{\Theta}(t)## is defined as
1698539743261.png

I want to understand the following calculation:
1698539807121.png

More specifically, the ##\text{Im}(G(\textbf{k},t)G(\textbf{k},-t))## from the first line to the second line. Everytime I write out the expression, the exponentials seem to just cancel and I get 0 for the imaginary part. Any assistance would be greatly appreciated.
 

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