of Cotes's Theorem. 21 



(X + X — 2 . COS. -^^ J. 



Let X + x"" = 2 . COS. c ; then x" + ^~" = 2 . cos. «^, and 



COS. fic — COS. a = 2""" (cos. ^ — cos. — ] (cos. c -^ 



COS. ^-I.LllzLJ!L\ ^ that is (by the formula cos. x — cos.jy = — 2 



. sin. — j^ . sm. -j=^l . sm. f--^ j . sin. -^-j = 2 



. sin. i[^ + c). sin. i ("-t^ + .) . &c. 



. sin. i (v - • sin. i (^' - c) . &c. 

 Let iti' = A + B, and ^^ = A — B, and by substitution, 

 the formula under consideration results. 

 This immediately gives the following 

 COS. ( A + B) . cos. ( A — B) = 2^"~^ • P • Q, vvhere 



P=sin. liii— j.sm. l-ii-;; / 



[(— 0.(^)-(A + B)l 

 sm. \ ;; J 



/(.„-.) (-^)-(A-B)l 



sm. \ !^ J 



and also, 



cos. (A + B) . sin. (A — B) = 2""""^ • P • Q, where 



P=:sin. llii-^ J . sm. 1 -^ J 



!(,„_,) (^)-(A + B)l 

 sin. [ ' — J 



Q = sm. -^) . sm. ( ^— J .... sm. ^ ;^ 



Hb) 



ric) 



