11–15. Identities Prove each identity using t...

Question

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

tanh(−x) = −tanh x

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says which statement about hyperbolic functions is valid? And we're given 4 possible choices as our answers. For choice A, we have the hyperbolic sign of minus Z is equal to the hyperbolic sign of Z. For choice B, we have the hyperbolic cosine of minus Z is equal to minus the hyperbolic cosine of Z. For choice C, we have the hyperbolic tangent of minus C as equal to minus the hyperbolic tangent of Z, and for choice D, we have the hyperbolic sine squared of Z, plus the hyperbolic cosine squared of Z is equal to 1. Now, before we look at each of these statements individually, let's recall our definitions for the hyperbolic sine, cosine, and tangent functions. So we call that for the hyperbolic sign. Of Z, that that is going to be equal to the quantity of E raised to the Z. Minus E raised to the minus Z in quantity, divided by 2. Also recall for the hyperbolic cosine function, we have the hyperbolic cosine of Z. And that's going to be equal to the quantity of E to the Z. Plus E to the minus Z in quantity, divided by 2. And finally, for the hyperbolic tangent function, we'll have the hyperbolic tangent of Z, it's going to be equal to. The hyperbolic sign of Z divided by the hyperbolic cosine of Z. So, with these definitions in mind, we'll now look at each one of these statements individually and evaluate whether they're true or false. So, just looking at choice A, on the left-hand side, we have the hyperbolic sign of minus Z. And so using our definitions, we'll rewrite this in terms of the exponentials. So, that means that the hyperbolic sign of minus C is going to be equal to the quantity of E to the minus C. Minus E to the Z. In quantity, divided by 2. And just rearranging this, this is going to be equal to, we'll factor out a minus sign out of the numerator, so we'll have minus the quantity of E to the Z minus E minus Z in quantity, divided by 2. Which is equal to minus the hyperbolic sign of Z. But if we look at the right hand side of the expression, for choice A, we see that we just have the hyperbolic sign of Z. That means that choice A is not correct, is a false statement. If we look at choice B on the left hand side of that equation, we have the hyperbolic cosine of minus Z. And again, writing this in terms of the exponentials, using our definitions, this is going to be equal to. The quantity of E to the minus Z plus E Z in quantity, divided by 2. Which is equal to. The hyperbolic cosine of Z. And if we compare that to the right hand side of the equation for choice B, we see that we have, in the answer choice, we have minus the hyperbolic cosine of Z, which is not what we uh actually calculated. That means that choice B is false. Now, if we look at choice C, on the left hand side, we have the hyperbolic tangent. Of minus Z. And using our definition, this is going to be equal to. The hyperbolic sign. Of minus Z. Divided by the hyperbolic cosine of minus Z. However, we just calculated that the hyperbolic sign of minus Z is equal to minus the hyperbolic sign of Z, and that the hyperbolic cosine of minus Z is equal to the hyperbolic cosine of Z. So making those two substitutions, we see that our hyperbolic tangent of minus Z is equal to minus the hyperbolic. Sign of Z. Divided by the hyperbolic cosine of Z. Which is going to be equal to by our definitions. Minus the hyperbolic tangent of Z. And if we look at our result here, compare it to the right-hand side of the equation for choice C, we see that that exactly matches what we calculated. So that means that choice C is actually a true statement. And finally, if we look at choice D here, We actually have an identity involving the hyperbolic cosine and hyperbolic sign functions. I recall that the identity involving the hyperbolic cosine and hyperbolic sign functions. Is that the hyperbolic cosine is squared of Z. Minus the hyperbolic sine squared of Z is equal to 1. And if we compare this identity to the identity given in choice D, we see that they are not equal. There is no way that I can rearrange our identity to equal that of choice D. That means that choice D is not correct. It is a false statement. That means that the only true statement was choice C. And so that is going to be the answer for this problem. And choice C says that the hyperbolic tangent of minus Z is equal to minus the hyperbolic tangent of Z. So I hope that this explanation has been useful, and I'll see you in the next video.

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

tanh(−x) = −tanh x

Key Concepts

Hyperbolic Functions

Odd and Even Functions

Proof Techniques in Calculus

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