
Jason T. answered 02/20/13
Jason Tutors Math & Physics
There are 2 trig identities which will help you with this problem:
1. cosh2(x) - sinh2(x) =1
2. cosh(2x) = cosh2(x) + sinh2x
Prove: sinh2(x) = (-1 + cosh(2x))/2
Let's work with the right hand side. First, let's rewrite it as (cosh(2x) - 1)/2
Use identity #2 above, to rewrite right side as ((cosh2(x) + sinh2(x)- 1)/2
From identity #1 we know that 1 = cosh2(x) - sinh2(x)
Substituting for 1 above the right hand side of the orginal becomes
(cosh2(x) + sinh2(x) - (cosh2(x) - sinh2(x))/2
Simplifying we get 2sinh2(x)/2 = sinh2(x) for the right side.
sinh2(x) = sinh2(x)