34 MATHEMATICS. TRIGONOMETRY. [KJCPOSBNTIAL EQD ATIOJCS. 



Binco by the last article _ 



- 2 n. 



w 

 Again 



and expanding each by the Binomial Theorem. 



, 2n.2T^t * , 2n.^=12n 2 



+ 2 ._ + . - T 2--. nr ,+ - 133- - 



2n.2u 1 * 2n.2n 



c , 2n. 2n 1 2n 2 2 s ,,, 



"-2li + ' Tas --- 87? 1 j | ' 



Now (a) and (6) are equal under all circumstances, and therefore their limiting values when 2 becomes very 

 large are equal, i.e., when ^ J; ---- ^ are each equal to zero. 



2 3 ,pir 1 



Now srnce _ . cotan.' = 



2n 



p*_ 

 2n 



But in the limit when 2n is very large |^ = 0. 



And the limit of ^; - when = is unity. 

 



Hence the limiting value of JMT""" *' 



And therefore the limiting value of ^ cotan. 1 F 



Hence the limiting value of the expression (o)^is 



2i 

 Again since (6) can be written 



( ** ** "\ 



The limiting value of (6) is 2* ^1 + JTJIJ + i. 2 .3.45 "*" ~ J 



Hence, , z\ f 

 or writing *' for **, we have 



Multiply both sides by *, and we obtain 



'"f^ + r^ =r1 ~" 1 ~ 



* 



- 



+ - and it. factors from formula (56), we can obtain cos. * in factor.. 

 Th, let *-! + 



__ 



2n.2n 1 2n 2 2n 3 

 . L2.3.4 



