By definition:
coshx =(e^x+e^-x)/2
so:
coshln(x+sqrt(x^2-1)) = 1/2(e^(ln(x+sqrt(x^2-1)))+e^(-ln(x+sqrt(x^2-1))))
coshln(x+sqrt(x^2-1)) = 1/2(x+sqrt(x^2-1)+1/(x+sqrt(x^2-1)))
coshln(x+sqrt(x^2-1)) = 1/2((x+sqrt(x^2-1))^2+1)/(x+sqrt(x^2-1))
coshln(x+sqrt(x^2-1)) = 1/2((x^2+2xsqrt(x^2-1)+x^2-1+1)/(x+sqrt(x^2-1)))
coshln(x+sqrt(x^2-1)) = 1/2((2x^2+2xsqrt(x^2-1))/(x+sqrt(x^2-1)))
coshln(x+sqrt(x^2-1)) =x(x+sqrt(x^2-1))/(x+sqrt(x^2-1))
coshln(x+sqrt(x^2-1)) =x