you mean √(1 + sinh^2(u)) = cosh(u) ? How, please?

Question

Need some clarification

Answer 1

By definition,

cosh^2(u) - sinh^2(u) = 1

Answer 2

sinh(x)=½(e^x-e^-x) and cosh(x)=½(e^x+e^-x) by definition of hyperbolic functions.

sinh²(x)=¼(e^2x-2+e^-2x) and cosh²(x)=¼(e^2x+2+e^-2x).

Therefore cosh²(x)-sinh²(x)=¼(e^2x+2+e^-2x-e^2x+2-e^-2x)=4/4=1.

Hence cosh²(x)=1+sinh²(x), cosh(x)=√(1+sinh²(x)).

Other hyperbolic identities follow by replacing the hyperbolic functions with their definitions.

For example, 2sinh²(x)=½(e^2x+e^-2x-2)=½(e^2x+e^-2x)-1=cosh(2x)-1, so:

cosh(2x)=1+2sinh²(x).