how do you find real and imaginary parts of cos(z)

Question : Find real and imaginary parts of cos(z)

Solution :

First we shall put \[z=x+iy\] then \[cos(z)=cos(x+iy)\]now using the formula of \[cos(A+B)=cos(A)cos(B)-sin(A)sin(B)\] we get \[cos(x+iy) = cos(x)cos(iy)-sin(x)sin(iy)\] now by the formulas \[cos(iy)=cosh(y) , sin(iy)=isinh(y)\] then we shall have \[cos(x+iy) = cos(x)cosh(y)-isin(x)sinh(y)\] now comparing it with a+ib form we get real part \[a = cos(x)cosh(y) \] and imaginary part \[b = -sin(x)sinh(y)\] are required real and imaginary parts of cos(z).