sinh2 x = (-1+cosh 2x)/2

Question

help me prove this please

Answer 1

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)

 

 

Answer 2

-1/2  +  (e^2x  +  e^-2x)/2*2  =  (e^2x   -2  +  e^-2x)/4  =  ((e^x  -  e^-x )/2)^2  =  (sinh x )^2

Answer 3

To prove an identity like this, it's convenient to work both sides and arrive at the same answer.

First the left hand side, using the definition of sinh:
sinh²x = ((e^x - e^-x)/2)² = (e^x - e^-x)²/4 = ((e^x)² - 2e^xe^-x + (e^-x)²)/4
= (e^2x + e^-2x - 2)/4

Now for the right hand side:
(-1 + cosh 2x)/2 = (-1 + (e^2x + e^-2x)/2)/2 = ((-2 + e^2x + e^-2x)/2)/2 =
(-2 + e^2x + e^-2x)/4 = (e^2x + e^-2x - 2)/4

So we arrive at the same result and the identity is proved! Make sure you understand each step...

Hope this helps!