Anglix — Trigonometrical Notes. 265 



_, / — . I sec a 



But cos a + v - I sm a = —= = =^ ; 



cos a - v - I sin a i - \/- i tan a 



therefore 



I X 



sec tti sec oo sec aj . . . sec a„ . :z=z . -■= . . . 



I - \/— I tan tti I - V - I tan ao 



= C0s(ai+ a2 + a-3+. . . +a„) + \/- i sin(ai + a2 + a3+. . . + a„), 



I -'s/- I tan a„ 

 that is, 



sec tti sec a^ sec a^ . . . sec a„ 1 1 - /^o + /?4 - . . . + \/- i {Ih ~ Ih + /jj - . . .) } 



= cos (oi + tto + ag + . . . + a„) + \/ - I sin (aj + a, + as + . . . + a„). 

 Equating possible and impossible parts, we get 

 sin (tti + tto + aj + . . . + a„) = sec ai sec tto sec a^ . . . sec a„ (/^i - h^ + h^ . . .), 

 cos (aj + ttj + 0.3 + . . . + .a,j) = sec ai sec ao sec aj . . . sec a„ ( i - ^o + ^^ - . . .), 



and therefore 



h-h + h- . . . ^ 



tan (^aj + a2 + as + . . . + a„j 



I - Ju + 7^4 - ... 00 



2. The following results may also be worthy of notice in connexion 

 with the foregoing. 



We have cos 7i6 + \/- i sin n9 = cos"6 ( i + v- 1 tan 6)". 



Expanding and equating possible and impossible parts, we get 



cos 7i9 . sec"9 = B, 



sin n9 . sec"9 = JD, 

 where 



B = 1 - Cn tan^^ + Ci tan*^ -■••■, 



the last term being 



n n — 1 



(- 1)2. tan"^, or (- I ) 2 ,n tan""^ 9, 

 according as n is even or odd ; and 



B = Cx tan 9 - Co tan^ 9 ^- c-^ tan^ 9 - . . . ., 

 the last term being 



n n — I 



n{- 1)2""' . tan"-i^, or (- i)^~. tan"^; 



for n even or odd ; and c,. denoting the number of combinations of n 

 things taken r at a time, •/. e. \j^ ^ |_^' | ''^ ~ ''• 



