The following definitions extend the sine and cosine functions to take arguments :
Prove the following formulas, where are complex numbers and .
- .
- .
- .
- .
- .
- .
- Proof. Using the given definition of the sine of complex numbers we have
- Proof. Similar to part (a) we compute
- Proof. We compute
- Proof. The two computations are as follows,
- Proof. We have,
- Proof. We have,