show that coth(x) -tanh(x)= 2 csch(2x)

Question

prove that this identity is true

Answer 1

coth(x) = 1/tanh(x) = ( e^x + e^-x)/( e^x - e^-x )

Therefore:

coth(x) - tanh(x) = ( e^x + e^-x)/( e^x - e^-x ) - ( e^x - e^-x)/( e^x + e^-x )

Add the 2 fractions and simplify.

It may help to use placeholder variables---in this case, recognize that the expression can be written as:

A/B - B/A

To simplify, we find the least common denominator, which is AB. We can therefore write (A² - B²)/AB

Any way you do it, the arithmetic is going to be tedious, so take it one step at a time....