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Prove the identity.

$ \cosh (-x) = \cosh x $ (This shows that $ cosh $ is an even function.)

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Missouri State University

Harvey Mudd College

Idaho State University

Okay, Now we are going to prove hyperbolic cosine or coach coach off. Negative X equals two kosher Becks, if you can recall, we know that if f off negative X equals two ffx, we know that f is called an even function. So this problem is equivalent to saying proof that Coach of X is an even function. Okay, like the previous problem, we're going to start with the definition. Coach of X is equal to e to the X plus e to the negative x divided by two. So to figure out what Coach of Negative X is, we simply substitute X with native X. This one's a lot simpler. You can see how things were going toe workout simply, the ex turns into negative X and the positive X Excuse me. The negative X turns into a positive X. So if you compare it to the top expression, you see that the left term and the right term simply just switched with each other. So this is exactly the same s koelsch X. So to sum it up, we start with the definition we substituted negative X and re proved that it's just the same as the original expression. And that proves that cosine hyperbolic cosine is an even function

University of California, Berkeley