I am stuck trying to show that $\displaystyle \int_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^x+1}=0$

I have tried using a Squeeze Theorem type approach, but at $\pi/4$ any function I choose overlaps and becomes less than or greater than the function depending. I am not sure where to start anymore. Anything will be helpful. Thank you in advance!

I am stuck trying to show that $\displaystyle \int\limits_{-\pi/2}^{\pi/2}\frac{\cos(2x)}{e^x+1}=0$

I have tried using a Squeeze Theorem type approach, but at $\pi/4$ any function I choose overlaps and becomes less than or greater than the function depending. I am not sure where to start anymore. Anything will be helpful. Thank you in advance!