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Prove the identity.$\coth^2 x - 1 = csch^2 x$

$\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x$

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Alex O.

June 21, 2020

thanks for skipping 20 steps,,,really helpful

Catherine R.

Missouri State University

Kayleah T.

Harvey Mudd College

Caleb E.

Baylor University

Boston College

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Video Transcript

So we want to prove this identity theater entity we're trying to prove is the co tangent H X squared minus one equals because she can a chicks, um, square. So we automatically recognize that co tangent is Thekla's scene over the sign. So this is what we're currently at aan den. We have minus one. We know that's going to be equal to based on our identities or properties that we know e the X plus Either the negative X squared over four over eat decks plus eating the negative eggs squared over four. And this is actually gonna be a minus because it's sign. So then this is minus one, which is equal to ive the X plus. Either the negative act squared over. Eat the X minus even negative X squared minus one. We can rewrite this now if we give it a common denominator. What we're left with is going to be either X plus either than negative X squared, um, minus e to the X minus e to the negative, x squared and then on the bottom. We'll have becoming denominator now. Then I'm distributing the squared through and simplifying what we end up being left with is going to be e to the two x plus two plus e to the negative two x minus e squared X plus two minus eating negative checks. So recognize that the each of the negative two x is will cancel. Um, so on also, the each of the two X will be canceled. So what we're left with is just four over our denominator of eating X minus E to the negative X square. So it's all just gonna be four. Uh, that is then going to be equal to two squared, which we can divide. So the way that we could view that is this is equal to one over either the x minus, eating the negative X squared over two squared. Um, And right there, what we end up having is that this is the same thing. Since if we got rid of this squared, this would be just sign the one over sign Age s. So what we have is one over sign age squared X, which is obviously equal to coast seeking h burn X