[...] 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. Aha, but
(c69) (sin(3*x)/sin(x),%%=trigcontract(factor(trigreduce(trigsimp(trigexpand(%%)))))) sin(3 x) pi pi (d69) -------- = 4 cos(x - --) cos(x + --) sin(x) 6 6 so prod((2*cos(t/3^k+%pi/4)+1)*(2*sin(t/3^k+%pi/4)-1),k,1,inf) = sqrt(2)*sin(t)+1 inf /===\ | | t pi t pi | | (2 cos(-- + --) + 1) (2 sin(-- + --) - 1) = sqrt(2) sin(t) + 1 | | k 4 k 4 k = 1 3 3 and prod((2*sin(t/(-3)^k-3*%pi/14)-1)*(2*sin(t/(-3)^k+%pi/14)-1)*(2*cos(t/(-3)^k+%pi/7)-1),k,1,inf) = -8*sin(t/2-3*%pi/7)*sin(t/2+%pi/7)*sin(t/2+2*%pi/7)/sqrt(7) inf /===\ | | t 3 pi t pi t pi | | (2 sin(----- - ----) - 1) (2 sin(----- + --) - 1) (2 cos(----- + --) - 1) | | k 14 k 14 k 7 k = 1 (-3) (-3) (-3) t 3 pi t pi t 2 pi 8 sin(- - ----) sin(- + --) sin(- + ----) 2 7 2 7 2 7 = - ----------------------------------------- sqrt(7) --rwg