Plane trigonometry, by S.L. Loney.

EXPANSIONS.

421

EXAMPLES. LXIII.
Expand in an infinite series
1+ a cos0 cos 0 - a cos (O - /)
1 2aco a 1 2a cos 4 a2 l-2cos '
sin 0-a sin (0-/)
3. - a sin ( - 4. eacos cos (0+ a sin ).
1- 2a cosb + a2 '
5. ea sin bO.
Prove that
a2 I
6. log - a2 in2 4 csin20 - c2sin220+ c3sin230 -a' cos2 0 + b sin2 L 1
where c = -
a+b
asin0. 1 1
7, tan- 1 a - a co si a sin 0 + 2 a3 sn 3...... adinf.
1 - a cos 0 2 3
I 1
8. tan-1 (sin a tan 2P) = sin a tan p + 3sin 3a tan3
+ sin 5a tan5 +.....ad inf.
9, If sin 0=x cos (0+a), expand 0 in a series of ascending powers
of x.
10. Expand y in terms of cos a, where
x+a x-a
2 tan y = sin x cosec -2 cosec 2
1-n
11. If tan x=n tany, and m=1- prove that
m2 7m3
x+r7r=y-msin2y+ -sin4y- 3 sin 6y +......ad inf.,
7r 7/
where r is to be so chosen that x + rr - y lies between - and +-.
2 2
12. What does the series of the preceding question become when
(1) n=cos a, and (2) n=-c?
cos 2a
13. Expand log cos (4 + ) in a series of sines and cosines of
ascending multiples of 0.