The Taylor series requires derivatives of the function sinh(z) in this problem. d/dz(sinh(z))=cosh(z) and d/dz(cosh(x))=sinh(z). So the odd numbered derivatives f', f''', f''''', etc., are all cosh(z), while all the even derivatives, including f itself are sinh(z). The hyperbolic functions are defined as sinh(z)=1/2(e^z-e^-z) and cosh(z)=1/2(e^z+e^-z). If z=a+ib, a complex number where a and b are real numbers, e^z=e^(a+ib)=e^a*e^ib=e^a(cosb+isinb). If z=0+ib (b would be an angle, usually expressed in radians) then e^z=cosb+isinb and e^-z=cosb-isinb, so sinh(z)=isinb and cosh(z)=cosb. The Taylor expansion is:
T(x)=f(z)+f'(z)(x-z)+f''(z)(x-z)^2/2!+f'''(z)(x-z)^3/3!+f''''(z)(x-z)^4/4!+...
So, substituting z=ib, T(x)=isinb+cosb(x-ib)+isinb(x-ib)^2/2+cosb(x-ib)^3/6+isinb(x-ib)^4/24+...=isinb(1+(x-ib)^2/2+(x-ib)^4/24+...)+cosb((x-ib)+(x-ib)^3/6+(x-ib)^5/120+...).
The series contains a mixture of real and complex terms, but the problem does not make clear what value b is, and this is the value on which the series is to be based. For example, if b=0, T(x) becomes x+x^3/3!+x^5/5!+... the standard series for the expansion of sinx.