Plane trigonometry, by S.L. Loney.

GREGORY'S SERIES. 397
341. When tan 0 is equal to unity, we have
log (1 + i tan 0) = log (1 + i) = log 1 + (cos - + i sin
so that the expansion of log (1 + i tan 0) is by Art. 336,
still true.
Similarly for log (1 - i tan 0).
Hence Gregory's series is true for the extreme values
0= - and 0= —.
4 4
r 37r
342. If 0 lie between r and -, or between
4 4
57T 77r
- and,......
4 4
or, generally, between
7r 37r
nr + - and nr+ 4,
tan 8 is greater than unity; in these cases the expansion
of log (1 + i tan 0) does not hold, and there is no such
expansion as equation (1) of Art. 340.
37r 5w
343. If 0 lie between - and, let it equal wr + a,
4 4
so that a lies between
-i and +
and hence, by equation (4) of Art. 340,
I 1
a = tan a - tan3 a + - tan5 a -......
3 5
But tan a = tan (0- r) = tan 0,
so that this equation is
- 7r = tan 0 - tan3 0+ tan5 0.......
d 5