CEUCES MATHEMATICS. 



19 



und the law of the coëfficieuts is established. 



.siu^=^_|;+^;- + 



(2/1—1)! 



The development ol' eos n may be etleeted in the same mauiier as that of sin 0, by 

 taking the relation •' 



cos 2f/-)-('os 2qj ■=-. 2 COH {1^ -[- <p) eiw {H — qi) ; 



hni il is more readily developed from the relation 



2 cos- ^/ = 1 + cos -111 (5). 



It is readily established that the series for cos 6 must have its first term unity and can 

 iuvoh'e only even iiovrers of (^. Therefore assume 



cos 14 = 1 -^a.,H- + rt,«' + (i,„-fi-"'- + 



Then, substituting in relation (H), 



2+2-a/- + 2'«^ft"+ .... T"a.:.fi-"+ .... 



2 ( 1 + 2a.fi"- -\-2a, \ H' + . . . . 2a,„ 



2a.,„.,a, 





H-" 



. . n odd ") 

 . . n even ) 



Equating coefficients of ff'", 



(2-" — 2-)a,„ = 2= f rt.,„ _.//, + ao„./(^+ a,i+ia„-\, « odd | 



t ia„a,„ n even j 



By equating coefficients of ^-, ^', Sec, we readily find that 



1 1 



2! 



a, = ^^ &c. 



Assume this law to hold up to the coefficient a.,,,.^ inclusive. Then a.,,,.-^^, «2„-i«i. &t'., 

 have all the same sign, and it is easily seen that this sign is given by ( — )". 



... (2--_i)«., = (-}ii^ (^^y- - + - 



^ ^ -" (2n) ! ( (2n — 2) ! 2 ! ^ (w 



(2«) ! . odd 



+ l)!(n — 1)! ' 

 (2«) ! 



! n !' 



;i ! n \ 



(—y (2'^"--— 1) by (3). 



(2n)! ^ ^ -^ ^ ^ 



. a. 



" (2n) ! ' 



