EXPONENTIAL EQUATIONS.] MATHEMATICS. TR IGONOMETRY. 633 



(57). To prove that 2* - ' sin. 2 ^ sin. 2 15 sin. 2 |T . . . . sin. 2 ^=^5 = 3 a. 



We have by actual division 



x" ' 

 Hence, when x = 1. The limiting value of r- will equal 1 + 1 + 1+ ____ + 1 + 1 to 2 n terms 2 n. 



X ^^ X 



Again by the last article 



?*LZl = (x + 1.) (x 2 2* cos. - + (*- 2xcos. ^ + lV... f x ! -2xcos. n ^ 

 x-1 'V n A / \ n 



x 2 " J 

 and hence when z = 1 , the limiting value of - _- 



is 



V n / V n 



and therefore these two limiting values are equal, and 



= 2 2 - 1 - 1 sin. 2 - sin. 8 1^ sin. 3 1^ ... . sin. 1 * gn ? . . . . (56). 

 In the same manner we may prove that 



2 = 2" sin. 1 -^ sin. 1 . sin. 1 . .... sin. 1 -^ . ... (57). 



4n 4n 4n 4n 



For 



TT 3*- , \ I , 2n IT. , ,"\ 

 ' 2x cos. 5- +lllz 2 2xcos. hlj I* 1 2* cos. ^ 1- 1 I 



Let z = 1, and tlieu observing that 



We shall obtain the required formula. 



(58). To express sin. x in a terto of factors. 



The above formula (53) for the factors of x 1 " a 1 " is true for all values of x and a 

 Suppose x = 1 + 7j"^ and a . 1 =- . 



Hence the factors on the right-hand side of Equation (53) will be equal to 



?f . 4 n. -^1 + -^ . cotan.' JLV .in.' ^ X 

 n 2n ^4n> 2 nf 2 n 



* * * 



2 cotan.' -.4Bin.' 



n 1^ 

 ^-- J 



in. ^ . . . . 4 sin.' 



.. . . . . . 



2n 2n 2n n 



-. 



2 n 2 n 



f 1 + A co^-'^l (l + A o^- J |^ V ( 1 + A 

 \ 4n 2 n/ \ 4n 2 2 nj \ 4n J 



cotan. 2 1 l + -? cotan.' j ... l + cotan. 

 4n 2 2 n/ \ 4n J li iy \ 4n a 2n 



VOL. i. 4 M 



