I'd use the definitions which extend the arguments to the complex case.
You mean the power series for for complex numbers. From here we can use the fact,Originally Posted by TD!
then you can easily denomstarte that,
But what about the other two facts, the necessary and suffienct conditions for the zero's of the sine function?
I was thinking, show that there exists a non-zero number on some interval which is a zero of the sine function by the use of the indetermediate value theorem. Then, demonstate that is is a zero then so is but that does not prove the "only if" part.