106 
MISCELLANEOUS EXAMPLES ON CHAPTER III 
41. Prove that 
sin?i0 = 2 ,l ~ 1 sin 0 sin ( d + ~ ) ... sin -¡0 + 
(«-1)' 
I- 
[Put x—a — 1 in Ex. XXY. 8, and change 0 into 20.] 
42. Prove that 
7r 2tr 7 TV . 1 
cos — cos ^ ... cos Ts = Ì28 * 
43. Prove that tan ~ tan ~ ... tan = 1, 
ATI ATI Alii 
44. Prove that 
(l+a?) n —(1 —#)" 
2;r 
=A ( ¿r 2 + tan 2 — ) ( x 2, +tan 2 
# 2 + tan 2 ), 
where A = 1, 1) if n is odd, and A=n, r — \n— 1 if % is even. 
45, If 1 j n ( x + tan 2 - j is expressed in the form 
2 A r { x+tan 2 
2» + l/ ’ 
% being a positive integer, show that 
( — l) r_1 2 . « 
A r = 
Oil! _ - LUO ,, , • 
2n + l 2« + l 2?i+l 
{Math. Trip. 1905.) 
[Apply the ordinary rule for partial fractions: it will be found that 
A r = {- l) r_1 2 sin 2 - 
rn » kir 
n cot 2 ; 
2w + l ‘ 2?i4-lfc=i 2?i + l’ 
and Ex. 40 can be used to obtain the given result.] 
46. Show that 
f(2r + l) 7T ] . 1N 
1 v — a }■ =71 COS {n— 1) a sec na. 
I 2« J 
{Math. Trip. 1907.) 
. (2r + l) n 
2 sin X—-—— cosec 
r=n 2 n 
[The right-hand side is 
l +x 
x n +x~ n ,r 2n + l ’ 
where x=cos a + i sin a — Cis a. The roots of x 2n + 1=0 are 
Gi^- 2 ^ (r=0, 1, 2n — \). 
Split up the right-hand side into partial fractions of the form 
/{*- 
. (2r+l)«- 
Cis 
2 n 
}• 
It will be found that A r = —i sin — Cis ^ —. To get the result 
All Ait 
in the form given we must associate the terms in pairs {r, n+r) where 
r = 0, 1, ..., «-!•]