PLANE TRIGONOMETRY. 83 



Then C + S^\ = 1 + 2z + 3;' +... 



- (n+ l)(cosn0 + N / lsiunG) + n{cos(n + 1)0 + N / 1 sini 

 t '2 sin -y J (cos + ^/ 1 sin 0) 



_ cosO-J-l sin0-(^4-l){cos(^-l)^+ N /-l sin^-^^j+yifcos?!^^-! sin 



Whence, equating possible and impossible parts, 

 c __ (n + 1) cos (n - 1) - cos - n cos nO 

 ^(l-cos6) ~' 



_ (n + 1) sin (n - 1) + sin - n sin n0 

 2(l-cos0) 



Many others may be summed by separating the r th term into 

 a <litr. -n 'in ut' tin- form #,+,- ^7 f , ^7 r being some function of r, 

 and U r+l the same function of r + 1. Thus to sum the s 

 111 1 



~ X T ,* "T* * ^k* "1* * I* * ._ A 



sin x sin 2aj sin 2 a sin2 'x 



we have 



1 _ riny'a; s 



sin 2 r sin S'' 1 sin 2'a; "" sin 2 r ~ > sin 2 r aj 



nee -: ocot -cota;, 



amx 



= cot x- cot : 



--cot2" t -cot2"- 1 *, 

 sm2^'x 



and the sum of the series is oot _ - cot 2"" 1 *. 



G 2 



