# Proving Hyperbolic Identity Using Osborn's Rule

• hex.halo
In summary, to find the hyperbolic equivalent of the trigonometric identity cos(x+y), we can use Osborn's rule to replace the sines with hyperbolic functions. To prove this identity, we can use the definitions of hyperbolic functions and algebraic manipulation, replacing cosh(x) with (
hex.halo

## Homework Statement

Given the trigonometric identity cos(x+y)... use Osborn's rule to write down the corresponding identity for cosh(x+y)... Use the definitionis of the hyperbolic functions to prove this identity

## The Attempt at a Solution

I can use Osborns rule to find the hyperbolic equivilent of the identity, however, I don't understand how I am to prove this identity...

$cos(x+y)=cosxcosy-sinxsiny$

Osborn said that when you have the product of two sines, you replace the sines with sinh and a negative sign.

so therefore $cosh(x+y)=cosxcosy-(-sinhxsinhy)$

rock.freak missed the h in his last bit for the 2 cos terms on the rhs.
$cosh(x+y)=coshxcoshy-(-sinhxsinhy)$

hex.halo said:

## Homework Statement

Given the trigonometric identity cos(x+y)... use Osborn's rule to write down the corresponding identity for cosh(x+y)... Use the definitionis of the hyperbolic functions to prove this identity

## The Attempt at a Solution

I can use Osborns rule to find the hyperbolic equivilent of the identity, however, I don't understand how I am to prove this identity...

Okay, you already know that cosh(x+ y)= cosh(x)cosh(y)+ sinh(x)sinh(y). Now replace cosh(x) by $(e^x+ e^{-x})/2$, replace sinh(x)= [itex](e^x- e{-x})/2[/tex], the corresponding things for cosh(y) and sinh(y) and do the algebra. What do you get when you multiply
$$\frac{e^x+ e^{-x}}{2}\frac{e^y+ e^{-y}}{2}$$

Last edited by a moderator:

## What is Osborn's Rule?

Osborn's Rule is a mathematical rule that states that the hyperbolic identity can be proven by taking the square of the cosine function and subtracting the square of the sine function.

## Why is it important to prove hyperbolic identity?

Proving hyperbolic identity is important because it allows us to better understand the relationship between hyperbolic functions and trigonometric functions, which has many applications in fields such as physics, engineering, and mathematics.

## Can Osborn's Rule be applied to all hyperbolic identities?

No, Osborn's Rule can only be applied to a specific set of hyperbolic identities, known as the hyperbolic Pythagorean identities.

## What are the steps involved in proving hyperbolic identity using Osborn's Rule?

The steps involved in proving hyperbolic identity using Osborn's Rule are: 1) Express the hyperbolic identity in terms of cosine and sine functions. 2) Square both sides of the equation. 3) Use Osborn's Rule to simplify the equation. 4) Rearrange the equation to show that both sides are equal. 5) Therefore, the hyperbolic identity is proven.

## Are there any limitations to using Osborn's Rule?

Yes, there are some limitations to using Osborn's Rule. It can only be applied to specific hyperbolic identities, and it may not work for more complex identities that cannot be expressed in terms of cosine and sine functions.

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