Hi @Andrew
At first, I thought you were describing the sum of arguments. This is certainly not a silly question. However, it requires some math skills (and patience) to express it for higher-order 
. Many years ago, I derived such an identity (using pen and paper), only to later discover that it is somewhat related to Chebyshev polynomials.
. Many years ago, I derived such an identity (using pen and paper), only to later discover that it is somewhat related to Chebyshev polynomials. syms x
%% 7th-degree
% y1 = (2^6)*cosh(x)^7
% y2 = cosh(7*x) + 7*(16*cosh(x)^5 - 8*cosh(x)^3 + cosh(x))
%% 5th-degree
% y3 = (2^4)*cosh(x)^5
% y4 = cosh(5*x) + 5*(cosh(3*x) + 2*cosh(x))
%% 3rd-degree
% y5 = (2^2)*cosh(x)^3
% y6 = cosh(3*x) + 3*cosh(x)
% isAlways(y1 == y2)
% isAlways(y3 == y4)
% isAlways(y5 == y6)
y7 = cosh(x)^7
y8 = (cosh(7*x) + 7*(16*(cosh(5*x) + 5*(cosh(3*x) + 2*cosh(x)))/(2^4) - 8*(cosh(3*x) + 3*cosh(x))/(2^2) + cosh(x)))/(2^6)
isAlways(y7 == y8)
Result:
