Simplify a product with f(x,n)?

Questions
1

Hello, I’ve a problem. Imagine you have a function f(x,n) like f(x,n)=x^n or sin(x*n). Then you have the product sign. The new formular is then a(x) = Product[f[x, n], {n, 0, x}]. And I don’t know how I can simplify that equation, or put that product sign away. Do you know how I can simplify an equation like that in general, or in specific Product[Cos[x/n], {n, 2, -1 + x}].

2

f[x_, n_] := x^n
a[x_] := Product[f[x, n], {n, 0, x}]
a
(* x^(1/2 x (1 + x)) )
a[2]
( 8 )
b[x_] := Product[Cos[x/n], {n, 2, -1 + x}]
b[5]
( Cos[5/4] Cos[5/3] Cos[5/2] *)

3

What exactly do you mean by “simplify”?

You may expand your product into a Sum. Is it that what you mean by simplify?

In[20]:= tt = Product[Cos[x[i]], {i, 1, 5}]

Out[20]= Cos[x[1]] Cos[x[2]] Cos[x[3]] Cos[x[4]] Cos[x[5]]


tt // TrigReduce

You will get a Sum of Cos[ x[1]+ rest ], the rest containing all the other x’s and all combinations of Plus and Minus.