Plane trigonometry, by S.L. Loney.

396

TRIGONOMETRY.

Hence (20 + 2n7r) i
=i tan 0 i 2 tan2 0 + i3 tan3 0......
- [-i tan- tian - t an 0-......
=-2 i i3 t a 0 + 3tan i 0 itan +......, + n7r =tan l- tan3 + 5 tan50-....(1),
where n is an integer.
The right-hand member of (1) may be written in the
two forms
tan 0 (1- tan2 ) +) tan 0 (1 - atan2 0 +...... (2),
and
tan 0-tan 0 tan2 - tan 0 1- tan2
3 \ 5 77 \ 9 /
+......(3).
If 0 lie between 0 and -, so that tan 0 is positive and
less than 1, then from (2) we see that the sum of the
series is positive, and from (3) that it is less than tan 0
and therefore less than unity,
In this case, therefore, n must be zero and we have
1 1
0 = tan - tan3 0 + - tan5 -...... ad inf....(4).
3 5
If we change the sign of 0, then every term in (4)
changes its sign, so that the series must also be true for
values of 0 between 0 and -
4