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How to derive the identity for hyperbolic cosecant?

The identity for hyperbolic cosecant is derived using the definition of hyperbolic sine and cosine.

Starting with the definition of hyperbolic sine:

sinh(x) = (e^x - e^-x)/2

We can rearrange this to solve for e^x:

e^x = 2sinh(x) + e^-x

Next, we can use the definition of hyperbolic cosine:

cosh(x) = (e^x + e^-x)/2

Substituting in the expression we just derived for e^x:

cosh(x) = (2sinh(x) + e^-x + e^-x)/2

Simplifying:

cosh(x) = sinh(x)/sinh(x) + cosh(x)

Multiplying both sides by sinh(x):

sinh(x)cosh(x) = sinh(x) + cosh(x)sinh(x)

Dividing both sides by sinh(x):

coth(x) = 1 + csch(x)

Finally, we can solve for csch(x):

csch(x) = 1/coth(x) - 1

Therefore, the identity for hyperbolic cosecant is:

csch(x) = 1/coth(x) - 1

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