Plane trigonometry, by S.L. Loney.

408 TRIGONOMETRY. [EXS. LX.]
1 1 1.3
15. 1+ cos 20 - cos40 + 2 cos 60-...... ad inf.
2 2.4 2. 4.6
n(n-1)
16. sinh u + n sinh 2u+ --- sinh3u +...... to n terms, where n is
a positive integer.
354. Ex. Sum the series
c2 cos 20 c4 cos 40
1+ 2 +...... adinf.
c2 cos20 c4 cos 4
Let C0 1+ +. c2 + -4 +........ad inf..(1),
12 ~1~
c2 sin 20 c4 sin 40
and S +...... ad inf....(2).
Hence
c2e20i &4e40i
C+Si= 1 + + + -4+...... ad inf.
where y = ce = c (cos 0 + i sin 0).
ey + e-Y.'. C+Si=
2
= ec cos +ic sin _- e-c cos0-csin 0.....(3
2 2(
= eccos0 [cos (c sin 0) + i sin (c sin 0)]
2
+ Z e-_co0 [cos (c sin 0)- z sin (c sin 0)]. (Art. 307.)
By equating real and imaginary parts we therefore
have
C = 1 cos (c sin 0) [e cos 0 + e-0 cos o]
= cos (c sin 0) cosh (c cos 0),