How do I prove that delta - sinc function is the same as an (-1)^n times the sinc $$\delta(n) - \frac{1}{2} \mbox{sinc} \left(\frac{n}{2}\right) = (-1)^n \frac{1}{2} \mbox{sinc} \left( \frac{n}{2} \right)$$

The picture shows what I've tried
Hint : Use the fact that, $\sin(-\frac{n\pi}{2}) = (-1)^n \sin(\frac{n\pi}{2}), \forall n \in \mathbb Z$