Question 1180100
{{{cosh(x)/(1+(sinh(x))^2 )= 2e^(x)/(1+e^(2x))}}}


by definition, two basic hyperbolic functions are


{{{sinh(x)=(e^x - e^(-x))/2}}} ->{{{sinh(x)=(e^x - 1/e^x)/2}}}->{{{sinh(x)=((e^(2x) - 1)/e^x)/2}}}->{{{sinh(x)=(e^(2x) - 1)/(2e^x)}}}


and


{{{cosh(x)=(e^x + e^(-x))/2}}}->{{{cosh(x)=(e^x + 1/e^(x))/2}}}->{{{cosh(x)=((e^(2x) + 1)/e^x)/2}}}->{{{cosh(x)=(e^(2x) + 1)/(2e^x)}}}



start with left side, substitute values above


={{{((e^(2x) + 1)/(2e^x))/(1+(e^(2x) - 1)/(2e^x)^2 )}}}



={{{((e^(2x) + 1)/(2e^x))/(((2e^x)^2+(e^(2x)- 1)^2)/(2e^x)^2 ))}}}---...simplify



={{{((e^(2x) + 1)/cross((2e^x)))/(((2e^x)^2+(e^(2x)- 1)^2)/(2e^x)^cross(2) ))}}}




={{{(e^(2x) + 1)/(((2e^x)^2+(e^(2x)- 1)^2)/(2e^x) ))}}}...simplify



={{{cross((e^(2x) + 1))/((e^(2 x) + 1)^cross(2)/(2e^x) ))}}}



={{{1/((e^(2 x) + 1)/(2e^x) )}}}



={{{(2e^x)/(e^(2 x) + 1 )}}}=>proven