- Prove that for
we have the following formulas,
- Using part (a) prove that
- Proof. We compute, using the definition of the complex exponential,
:
(Where in the second to last line we used that cosine is an even function and sine is odd, i.e.,
and
.)
For the second formula we compute similarly, - Proof. We can compute these directly using the expressions we obtained in part (a),