16–18. Identities Use the given identity to p...

Question

16–18. Identities Use the given identity to prove the related identity.

Use the identity cosh 2x = cosh²x + sinh²x to prove the identities cosh²x = (cosh 2x + 1)/2 and sinh²x = (cosh 2x − 1)/2.

Accepted Answer

Welcome back, everyone. In this problem, using the definitions of hyperbolic functions, which of the following expressions is equivalent to cash squared Z minus cinch squared Z? A says 0, B1, C2, and D synch2 Z. Now, what are the definitions of hyperbolic functions that can help us to solve this problem? Well, recall. That Kai. is equal to ez plus e to the negative z divided by 2. And we also know that since the. Is equal to e minus ez divided by 2. So if we can use both of these in our expression, cash squared Z minus sci squarez, then we should be able to simplify and figure out which of these options is correct. So now, by using our definitions, that means we could rewrite our expression, OK. As E Z plus e negative z divided by 2 squared for cash squared z minus E Z minus e to the negative z divided by 2 square for cinch squaredz. OK, now if we square both our terms for the first term it is going to become e to the power of 2 z + 2 + e to the negative 2 Z. OK, all divided by 4. And for a second term, it's gonna be E to the power of 2 Z minus 2, OK, plus E to the power of negative 2 all divided by 4. Notice both of them have a common denominator of 4, so we can combine our numerators. So that's E 2 Z + 2 plus E to the negative 2 Z. OK. Minus our entire expression here. E 2 Z minus 2 plus E to the negative 2 Z. OK. I know if we distribute our negative sign. Then here, let me just make some changes. This is going to become -2 minus E to the 2 Z plus 2, because we're subtracting negative 2 minus, OK, E to the negative 2 Z. OK? So those are how our signs will change. Now what do you notice? Well, here, e 2 Z minus minus 2 is 02 Z minus e to the negative2z is also 0, but we are left with 2 + 2, which equals 4 and 4 divided by 4 equals 1. Therefore, that tells us that cash squared z minus sci squarez is equivalent to 1. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

16–18. Identities Use the given identity to prove the related identity.

Use the identity cosh 2x = cosh²x + sinh²x to prove the identities cosh²x = (cosh 2x + 1)/2 and sinh²x = (cosh 2x − 1)/2.

Key Concepts

Hyperbolic Functions

Hyperbolic Identities

Algebraic Manipulation

Watch next

16–18. Identities Use the given identity to prove the related identity.Use the identity cosh 2x = cosh²x + sinh²x to prove the identities cosh²x = (cosh 2x + 1)/2 and sinh²x = (cosh 2x − 1)/2.