Anglin — Trigonometrical Notes. 263 



"We have 



( I - ffi xy^ ( I - a-i xy^ ( I - ^3^^;)"^ ... (1 - a,^xy^ 



= I -\- h]^x + hn x~ + Ji^x^ + . . . + h,. aj*" I . . . 00 . 

 Putting X = ± -y- I in succession, we find 



(i _«^ V^y{i - a, y~iy (i-a, y^y . . . (i - «„ y~) 



= i-h + hi- . . .+ V - I ijh - h + /«5 - . • '), 

 and 



( I + ffl a/- I )-' ( I + ffo v/'- I )-' ( I + ^3 y/- I )-^ . . ( I + r?„x/- I )-^ 



= I - h. + Jh- . . . - v/- 1 (Ai - 7^3 + ^5 _ . . .). 



Thus ^-h,-h-.h-...^ 



I - 7«2 -f- Ai - . . . OO 



_(i +giV-i)(r+g2\/^)...(i+g„V-i)-(i-giV-i)(r-ff.2V-r)... (r-ff«V^) 



(l +«iV^)(l+«2\/^)... (l+«„\/^) + (l-fflV-l)(l-«2V- l) ... (l_ff„V^) 



Again, 



( I + aix){ I + a2x){ I + f/3^) . . . ( I + a„,x) = i + CiX + c^x^ + . . .-\- c,^ x", 

 where e,. denotes the sum of the products of a^, do, a-^, ... «,„ taken r 

 at a time. Putting x = ± v - i in succession, we find 



( I + «i \/- I )( I + «o \/- I )( I + «3 v^- I ) • • • ( I + *u v^- J ) 



= I - Co + (?4 - . . . + \/- I {c^ - Co + c-^ - . . .), 



and ( I - rti V - I )( I - <ro\/ - I ) ( I - ff3 \/- i ) . . . ( i - ff,. \/- i ) 



= I - C2 + C4- . . . - -\/- I (ci - C3 + C5 - . . .). 



^ h^-h+lh- . . .OO Ci - (-3 + C5 - . . . 



Hence 



I - A2 + /^4 - . . . W I - (3 + C'i - . . . 



(the last terms in the numerator and denominator of the right-hand 



u 11 n — \ 



member being (- l)^"'. c„_i and (- 1)2 . c,„ or (- i) 2 . c„ and 



11 — z. 



(- i)~2~. r„-,, according as n is even or odd respectively) ; and there- 

 fore 



_, ^'1 -l^Z + l>r,- • • • ^ 



tan~'ffi + tan^'^o + taur^ci. + . . . + tan^' a„ = tan ; j . 



