why only the number 2.
consider the following relation where x and y are functions of a given variable: x^n+y^n=1; if we drift, we get the differential equation: nx'x^(n-1)+ny'y^(n-1)=0 --- x'x^(n-1)+y'y^(n-1)=0 This equation has for solution: x'=-y^(n-1) et y'=x^(n-1) for n=2; x^2+y^2=1 x'=-y et y'=x we deduce: x=cos(x) et y=sin(x) for n=4; x^4+y^4=1 x'=-y^3 et y'=x^3 I defined by : x=qos(x) et y=qin(x) qos(x)=sum(a_(n)*(-1)^(n)*x^(4*n)/(4*n)!,n=0..infinity) qin(x)=sum(b_(n)*(-1)^(n)*x^(4*n+1)/(4*n+1)!,n=0..infinity) 1) why all mathematics seems to rest on the index n = 2; 2) I could not simplify the coefficients a_ (n) and b_ (n) despite all my efforts ... Can we put together a simplified algorithm? 3) for n = 0, what does the x and y functions return?