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Is this what you need ?

sage: ex2 = ex.subs_expr(sinh(x/2) == (exp(x/2)-exp(-x/2))/2)
sage: ex2.exp_simplify()
e^x - 1

Is this what you need ?

sage: ex2 = ex.subs_expr(sinh(x/2) == (exp(x/2)-exp(-x/2))/2)
sage: ex2.exp_simplify()
e^x - 1

or even better

sage: w0 = SR.wild(0)
sage: ex2 = ex.subs_expr(sinh(w0) == (exp(w0)-exp(-w0))/2)
sage: ex2.exp_simplify()
e^x - 1