Proving the Cosh and Sinh Identity

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In summary, the conversation discusses an equation involving exponential, hyperbolic sine, and hyperbolic cosine functions. The person is struggling to show that the equation is correct for an exercise and has tried using the relation cosh^2-sinh^2=1. Another person suggests using the definition of sinh and cosh to express everything as exponentials and the correct relation cosh^2-sinh^2=1.
  • #1
aaaa202
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Is it true that:

exp(2x)sinh(y)2 + exp(-2x) = exp(2x)cosh(y)2-2sinh(2x)

I need this to be correct for an exercise but I don't know how to show it. I tried using something like cosh2+sinh2=1, but it didn't work.
 
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  • #2
Perhaps use the definition of sinh and cosh to express everything as exponentials?
 
  • #3
aaaa202 said:
I tried using something like cosh2+sinh2=1, but it didn't work.

In general cosh2+sinh2 ≠1 :wink:
 
  • #4
aaaa202 said:
Is it true that:

exp(2x)sinh(y)2 + exp(-2x) = exp(2x)cosh(y)2-2sinh(2x)

I need this to be correct for an exercise but I don't know how to show it. I tried using something like cosh2+sinh2=1, but it didn't work.

It might also help if you use the correct relation. cosh^2-sinh^2=1.
 
  • #5
I thought ;) would propel him there. ;)
 
  • #6
epenguin said:
I thought ;) would propel him there. ;)

Yeah, I didn't see your hint until after I posted. Sorry.
 
  • #7
No matter. :smile:
 

FAQ: Proving the Cosh and Sinh Identity

What is the cosh and sinh identity?

The cosh and sinh identity is a mathematical relationship between the hyperbolic cosine (cosh) and hyperbolic sine (sinh) functions. It states that cosh2(x) - sinh2(x) = 1.

Why is the cosh and sinh identity important?

The cosh and sinh identity is important in many areas of mathematics and physics, particularly in the study of hyperbolic functions and their applications. It also has applications in areas such as differential equations, complex analysis, and special relativity.

How is the cosh and sinh identity derived?

The cosh and sinh identity can be derived using the definitions of cosh and sinh in terms of exponential functions. By substituting these definitions into the identity and using basic algebraic manipulations, the identity can be proven.

What are some common uses of the cosh and sinh identity?

The cosh and sinh identity is commonly used in solving differential equations involving hyperbolic functions, as well as in simplifying and solving trigonometric equations. It also has applications in areas such as signal processing and electrical engineering.

Are there any variations of the cosh and sinh identity?

Yes, there are several variations of the cosh and sinh identity, including the double angle identity, the half angle identity, and the inverse identity. These variations involve different combinations of cosh and sinh functions and can be useful in different mathematical applications.

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