GENERAL THEORY 
199 
since tana: is an odd function. From 1), 2) we have, clearing 
fractions, 
a F> I. “l J I 
4 ! 6 ! 8 \) 
, ( a K , a- a-.\ 7 , / a 7 
+ r ,_ 2l x 
Comparing coefficients on each side of this equation gives 
a x — 1. 
a x _ 1 
<*3 9 ? 
U/Q 
*5 “ 7m 
Wrr 
«9-^ 
3! 
a, 1 
a, = 
3 8 
5!' 
J? 
II 
£1' 
a x 
1 
~ 6 ! _ 
7!' " 07 
a. 
1 
6! 
8! 
9 ! 9 
17_ 
315' 
62 
2835' 
Thus t, an j;=3: + |^3 + t 2 5* 6 + X 7 + 2! f 2:9 + 
(3 
vafo’d in ( — 
2 2 
Example 2. 
fix') — cosec x ■■ 
sm X 
1 
•"-iHl-v 
\ xP x(l— Q) 
Since 
we see that 
Q = 1- 
sin 2: 
K?l <1 
when a: is in 33 = ( — 7r + 8, 7r — S), S > 0. Thus xf (x) = P 
can be developed in a power series in 33- As f(x) is an odd 
function, xf(x) is even, hence its development contains only even 
powers of x. Thus we have 
xf(x) = a 0 + a 2 a: 2 + a 4 a; 4 + •••