Converting to "standard form" (the quotient for two values of t recovers the finite version), prod(8*cos((t/(2^k))-((3*%pi)/7))*sin((t/(2^k))-((5*%pi)/14))*sin((t/(2^k))-((3*%pi)/14)),k,1,inf) =-((8*sin(t-((3*%pi)/7))*cos(t-((5*%pi)/14))*cos(t-((3*%pi)/14)))/(sqrt(7))) inf /===\ | | t 3 pi t 5 pi t 3 pi | | 8 cos(-- - ----) sin(-- - ----) sin(-- - ----) | | k 7 k 14 k 14 k = 1 2 2 2 3 pi 5 pi 3 pi 8 sin(t - ----) cos(t - ----) cos(t - ----) 7 14 14 = - ------------------------------------------- . sqrt(7) Now it looks obvious, and it sort of is. If we rewrite as Prod cos cos cos = sin sin sin, then we have cos cos cos = (sin 2) (sin 2) (sin 2)/sin sin sin and the phases simply permute when doubled mod 2pi. Analogously prod(8*cos(x/(-2)^k-2*%pi/9)*cos(x/(-2)^k+%pi/9)*cos(x/(-2)^k+4*%pi/9),k,1,inf) = -8*sin(x-2*%pi/9)*sin(x+%pi/9)*sin(x+4*%pi/9)/sqrt(3) inf /===\ | | x 2 pi x pi x 4 pi | | 8 cos(------ - ----) cos(------ + --) cos(------ + ----) | | k 9 k 9 k 9 k = 1 (- 2) (- 2) (- 2) 2 pi pi 4 pi 8 sin(x - ----) sin(x + --) sin(x + ----) 9 9 9 = - ----------------------------------------- sqrt(3) And, of course prod(32*sin(t/(-2)^n-5*%pi/22)*cos(t/(-2)^n-2*%pi/11)*sin(t/(-2)^n-%pi/22)*cos(t/(-2)^n+%pi/11)*cos(t/(-2)^n+4*%pi/11),n,1,inf) = -32*cos(t-5*%pi/22)*sin(t-2*%pi/11)*cos(t-%pi/22)*sin(t+%pi/11)*sin(t+4*%pi/11)/sqrt(11) inf /===\ | | t 5 pi t 2 pi t pi t pi | | 32 sin(------ - ----) cos(------ - ----) sin(------ - --) cos(------ + --) | | n 22 n 11 n 22 n 11 n = 1 (- 2) (- 2) (- 2) (- 2) t 4 pi cos(------ + ----) n 11 (- 2) 5 pi 2 pi pi pi 4 pi 32 cos(t - ----) sin(t - ----) cos(t - --) sin(t + --) sin(t + ----) 22 11 22 11 11 = - --------------------------------------------------------------------. sqrt(11) Ooh, this is fun. prod(16*sin(t/2^n+%pi/30)*cos(t/2^n+%pi/15)*cos(t/2^n+2*%pi/15)*cos(t/2^n+4*%pi/15),n,1,inf) = 16*cos(t+%pi/30)*sin(t+%pi/15)*sin(t+2*%pi/15)*sin(t+4*%pi/15) inf /===\ | | t pi t pi t 2 pi t 4 pi | | 16 sin(-- + --) cos(-- + --) cos(-- + ----) cos(-- + ----) | | n 30 n 15 n 15 n 15 n = 1 2 2 2 2 pi pi 2 pi 4 pi = 16 cos(t + --) sin(t + --) sin(t + ----) sin(t + ----), 30 15 15 15 prod(-16*cos(t/(-2)^n-2*%pi/15)*sin(t/(-2)^n-%pi/30)*cos(t/(-2)^n+%pi/15)*cos(t/(-2)^n+4*%pi/15),n,1,inf) = -16*sin(t-2*%pi/15)*cos(t-%pi/30)*sin(t+%pi/15)*sin(t+4*%pi/15) inf /===\ | | t 2 pi t pi t pi t 4 pi | | - 16 cos(----- - ----) sin(----- - --) cos(----- + --) cos(----- + ----) | | n 15 n 30 n 15 n 15 n = 1 (-2) (-2) (-2) (-2) 2 pi pi pi 4 pi = - 16 sin(t - ----) cos(t - --) sin(t + --) sin(t + ----) 15 30 15 15 But this isn't the whole story, as it won't produce a naked t on the rhs (which we've seen), nor products with (+-3)^n, e.g. --rwg SOCIAL BRETHREN BRONCHIAL TREES CAMPHOR TREE PERCHROMATE UNRESTRAINED SATURNINE RED TRAINED NURSE