Prove sinh2x=2sinhx*coshx

We'll write the formula for each term of the given
identity:

sinh2x = (1/2)(e^2x -
e^-2x)

sinh x = (1/2)(e^x -
e^-x)

cosh x =  (1/2)(e^x +
e^-x)

We'll re-write the identity that has to be
demonstrated:

(1/2)(e^2x - e^-2x) = 2*(1/2)*(1/2)(e^x -
e^-x)*(e^x + e^-x)

We'll simplify and we'll
get:

(1/2)(e^2x - e^-2x) = (1/2)(e^x - e^-x)*(e^x +
e^-x)

We'll re-write the product from the right side as a
difference of squares:

(1/2)(e^2x - e^-2x) = (1/2)[(e^x)^2
- (e^-x)^2]

We'll multiply the exponents and we'll
get:

(1/2)(e^2x - e^-2x) = (1/2)(e^2x -
e^-2x) q.e.d.