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# Prove the identity. $\sinh (-x) = -\sinh x$ (This shows that $\sinh$ is an odd function.)

## $-\sinh x$

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### Video Transcript

All right, So in this problem, we are going to prove sinche. The hyperbolic Sinus sometimes pronounce inch. So for simplicity, I'm going to call it that way. We're going to prove cinch off. Negative X is negative. Sin jacks. Now, if you can recall that f off negative, X equaling effort negative ffx when this happens, F of X is called an odd function. So this problem is equivalent to saying, since of X is an odd function, and that's what we're going to prove. All right, so we start the proof with the definition. What? It's sinche FX. We know that this is equal to e to the X minus e to the negative x divided by two. So to figure out what cinch off negative X is equal to, we substitute X with negative X and see what's gonna happen. The first X becomes a negative x negative X becomes a positive X, And as you can see, you can pull out a negative from both of these two terms on the numerator. So it'll be negative e to the x minus into the matrix, all divided by two. And we know that this portion is cinch of X, so it to sum it up, this is negative cinch of X, and that is exactly what we're trying to prove. So to summarize it, we start with the definition we plug in Negative X and then showed that we are able to pull out a negative, and that proves that cinch of X is an odd function.

University of California, Berkeley

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