Plane trigonometry, by S.L. Loney.

COMPLEX QUANTITIES. EXPONENTIAL SERIES. 365
When x is complex, ex stands for a series of the same form
as that series which, when x is real, has been proved
to be equal to
+1+ Ai +......
Instead of ex the expressions E (x) and exp (x) are sometimes used.
305. By a proof similar to that of Art. 300, C.
Smith's Algebra, it may be shewn that
ex, ey = ex+,
whether x and y be real or complex quantities, so that
the functions el and ey obey a law of the same form as
the index law.
306. If x be put equal to Oi, where 0 is real, we
then have
92j2 5j33
e 1- ^+ - +......
02 s4 06
8 + -. -.......
= cos 0 + i sin 0. (Arts. 279 and 280.)
So e-6i = cos 0 - i sin 0.
Hence, by addition, we have
eOi + e-Oi
cos 0= 2
and, by subtraction,
ei e-e2 i
sin 0=-
2i '