How to derive the identity for hyperbolic cosecant?

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