Anglin — Ti-igonometrkal Notes. 261 



where a-^, a^, a-i, • ■ ■ «,„ «„+i are roots of 



x"^^ - fi x" + f-z x"-^ - t^ x"^' + . . . + t„x"- - t„^i = o. 

 (2) If n be odd, the last term in the numerator above referred to 



n — i n—T. 



is (- i) 2 . c^^^ and the last term in the denominator is (- i) 2 . c^^x- 

 Hence, proceeding as before, we shall get 



C 

 tan"' ffi + tan"' «., + tan"' %+... + tan"' «„ + tan"' (7„+i = tan"' ■- , 



where 



« — I 



C= (ci + rt„^i) - (C3 + c. «„+i) + (C5 + c^ ff.„+i) -... + (- 1 ) 2 . (<?„ + c„.i . ff„^.i), 



and 



11 — I 

 Z) = I - (c, + Ci «„+i) + {c^ + C3 ff,,+i) -... + (-1)2 . ((,^^_^ + c^^_, ,. a,^^^ 



n-\- \ 

 + (-1)2 . C,, ff„+i. 



But ffi, «,, «3, . . . a„, ff„+i are roots of 



«"+' - (ci + a„+i) a.'" + (c, + Ci «„+i) :»"-' - (<?3 + Co ff,,+i) x''-"- + . . . 



Hence, since n is odd, 



tan"' «! + tan"' «., + tan"' ^3 + . . . + tan"' «„ + tan"' «,j^i 



?z — I 



= tan"' ^j^ , 



l-^,.+ ^,-... + (-l) 2 .t,,,^ 



where «i, %, ^^3, ... «„, ff„+i, are roots of 



Thus the proposition is completely established. 

 Cor. — Putting (ri = tanai, «o = tana2, ffj = tan as, ...«„ = tan a„, it 

 follows that 



, . ^ fi - ^3 + <7s - . . . 



tan (_ai 4- Oo + 03 + . . . + a,J 



1 - Co + f 4 - . . . 



where c,. denotes the sum of the products of tau aj, tan a,, tan 03, . . . 

 tana,,, taken r at a time — the well-known trigonometrical formula, 

 which is thus established without the use of the symbol %/ - 1. 



