Hello, just to let ... product(cos(theta/5^k)-4*(sin(theta/(4*5^k)))^2*cos(theta/(2*5^k)), k, 1, inf)=cos(theta/4); prod((cos(theta/5^(k))+4*(cos(theta/(4*5^(k))))^2*cos(theta/(2*5^(k))))/5,k,1,inf)=(4/(theta))*sin(theta/4); product(cos(theta/5^(k/2))-4*(sin(theta/(4*5^(k/2))))^2*cos(theta/(2*5^(k/2))), k, 1, inf)=cos(theta/4)*cos(theta*sqrt(5)/4); product(cos(theta/5^(k/3))-4*(sin(theta/(4*5^(k/3))))^2*cos(theta/(2*5^(k/3))), k, 1, inf)=cos(theta/4)*cos(theta*5^(1/3)/4)*cos(theta*5^(2/3)/4); product(cos(theta/5^(k/7))-4*(sin(theta/(4*5^(k/7))))^2*cos(theta/(2*5^(k/7))), k, 1, inf)=cos(theta/4)*cos(theta*5^(1/7)/4)*cos(theta*5^(2/7)/4)*cos(theta*5^(3/7)/4)*cos(theta*5^(4/7)/4)*cos(theta*5^(5/7)/4)*cos(theta*5^(6/7)/4); product(cos(theta/5^(k/m))-4*(sin(theta/(4*5^(k/m))))^2*cos(theta/(2*5^(k/m))), k, 1, inf)=prod(cos(5^(i/m)*theta/4),i,0,m-1); prod((cos(theta/5^(k/m))+4*(cos(theta/(4*5^(k/m))))^2*cos(theta/(2*5^(k/m))))/5,k,1,inf)=(4^m/(theta)^m)*prod(sin(5^(i/m)*theta/4)/5^(i/m),i,0,m-1); Euler's identity extension: product(cos(theta/2^(n/m)),n,1,inf)=product(sin(2^(i/m)*theta)/2^(i/m),i,0,m-1)/theta^m; with hyperbolic functions: product(cosh(theta/2^n),n,1,inf)=sinh(theta)/theta; product(cosh(theta/2^(n/m)),n,1,inf)=product(sinh(2^(i/m)*theta)/2^(i/m),i,0,m-1)/theta^m; prod(2*cosh(theta/2^(n/m))-1,n,1,inf)=prod((1+2*cosh(2^(i/m)*theta))/3,i,0,m-1); prod((2*cosh(theta/3^(n/m))+1)/3,n,1,inf)=(2^m/(theta)^m)*prod(sinh(3^(i/m)*theta/2)/3^(i/m),i,0,m-1); prod((cosh(theta/5^(k/m))+4*(cosh(theta/(4*5^(k/m))))^2*cosh(theta/(2*5^(k/m))))/5,k,1,inf)=(4^m/(theta)^m)*prod(sinh(5^(i/m)*theta/4)/5^(i/m),i,0,m-1); Best regards