# Cosh(2z) Equals Cosh^2(z) Plus Sinh^2(z)

• Wardlaw
In summary, cosh(2z)=cosh^2(z)+sinh^2(z) is a hyperbolic function identity that can be shown by comparing it to the similar identity for circular trigonometric functions, cos(2z)=cos^2(z)-sin^2(z). By considering z as a random variable, it can be solved by manipulating the expressions on the right hand side and using standard forms. The solution involves finding the product of two sines and flipping the sign when converting to hyperbolic functions.
Wardlaw
Show that cosh(2z)=cosh^2(z)+sinh^2(z)

?

Hmm, I was just considering z as a random variable label, could just as easily be a, theta, or x.

So comparing to cos(2z) == cos2z - sin2z, there is a product of 2 sines, which you flip the sign of when comparing to hyperbolics, so cosh(2z) == cosh2z + sinh2z

Welcome to PF!

Hi Wardlaw!

(try using the X2 tag just above the Reply box )
Wardlaw said:
Show that cosh(2z)=cosh^2(z)+sinh^2(z)

?
Wardlaw said:
Yeah. I tried using the standard form for these expressions, when considering the RHS. I am then left with a quarter e^2z. Could you check this please?

You should get some e-2z also.

Show us what you got for the RHS.

tiny-tim said:
Hi Wardlaw!

(try using the X2 tag just above the Reply box )

You should get some e-2z also.

Show us what you got for the RHS.

Oh yeah you are correct, my mistake. I can't even read my own working :)
How exactly do you go about solving thi problem?

Wardlaw said:
How exactly do you go about solving thi problem?

I leave it to you.

tiny-tim said:
I leave it to you.

Solved

Woohoo!

## What is the equation "Cosh(2z) Equals Cosh^2(z) Plus Sinh^2(z)"?

The equation states that the hyperbolic cosine of twice the variable z is equal to the sum of the square of the hyperbolic cosine of z and the square of the hyperbolic sine of z.

## What is the significance of the equation "Cosh(2z) Equals Cosh^2(z) Plus Sinh^2(z)"?

This equation is a fundamental relationship in hyperbolic trigonometry and is often used in solving problems involving hyperbolic functions.

## How is the equation "Cosh(2z) Equals Cosh^2(z) Plus Sinh^2(z)" derived?

The equation can be derived using the double angle formula for the hyperbolic cosine and the fundamental identity for hyperbolic functions.

## What are the applications of the equation "Cosh(2z) Equals Cosh^2(z) Plus Sinh^2(z)"?

The equation has various applications in fields such as physics, engineering, and mathematics. It is used in solving problems related to hyperbolic functions, such as calculating the area under a hyperbolic curve or finding the solution to differential equations.

## Are there any other equivalent forms of the equation "Cosh(2z) Equals Cosh^2(z) Plus Sinh^2(z)"?

Yes, the equation can also be written as cosh(2z) = 1 + 2sinh^2(z) or cosh(z)^2 - sinh(z)^2 = 1. These are all equivalent forms of the same fundamental relationship between hyperbolic functions.

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