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You might want to familiarize with the Maxima and SymPy parts of Sage. With Maxima you can do
sage: ex = cosh(x) - (exp(x)+exp(-x))/2
sage: ex._maxima_().exponentialize().sage()
0
Here the Maxima function exponentialize does the rewrite. In Sympy:
sage: ex = cosh(x) - (exp(x)+exp(-x))/2
sage: import sympy
sage: sympy.simplify(ex)
0
