sink@, cosk@ are expanded in ascending powers of sin®@. 245 
ke? (? ale ae) 
=1- sin? + hi 
erate (c) Bae 
ke (k? — 2?) ... {k® —(2n — 2°} 
el 2n! 
x sin 6 — ee | 
2. Similarly let 
9 
Un =) (9 —t) sin” tdt, 
0 
and integrate by parts. Thus 
p 
Wy = | n(@ — t)sin™"¢.sin tdt 
0 
9 
=— c (@ — t) sin” ¢ cos | 
0 
0 
+n | {(n— 1)(@ —t) sin” ¢ cos? ¢ — sin” ¢ cos t}dt 
0 
9 
=— c (@—t) sin” ¢ cost + sin” | 
0 
8 . 
+n(n— 1) (9 —t) sin” ¢ cos? tdt 
0 
=—sin”6+n(n—1) (Uns — Un), 
Z 1! ; n 
ce n 
amd VU, > Web) sin” @ + eae 
This holds when n= 2, 3, 4..., but when n=1, 0 we have 
v%,=O—-—snd, y=46%. 
Hence 6=sin 60+, 
ism? 1.3 sim’. 
=sin@ +5 Sealer aa ne 
13 2 (2n = 1) [isin 2 
: aise le woees 3 
78.4... 2n = + (2041) tina (3), 
Get 
and 59 = 5 sin 6 + 2v, 
_ sin? 6 2sint@ | 2.4 
oer sea 8 
