SOLUTION: prove the identity: a) cosh x - sinh x = e^-x b) find the derivative f(t) = csch t(1- ln csch t)

Algebra ->  Proofs -> SOLUTION: prove the identity: a) cosh x - sinh x = e^-x b) find the derivative f(t) = csch t(1- ln csch t)      Log On


   

Question 912958: prove the identity:
a)
cosh x - sinh x = e^-x
b)
find the derivative
f(t) = csch t(1- ln csch t)
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!

re TY
Yes, practice is good.
Recommend carefully keeping things 'intact' until You go in 'for the finish"
a)
coshx = (e^x + e^(-x))/2
-sinhx = -(e^x - e^(-x))/2)
cosh x - sinh x = 2(e^(-x))/2) = e^(-x )
............
b)f(t) = csch t(1- ln csch t)
f' = -csch(t)coth (t) - [-csch(t)coth (x)ln(csch(t) + csch(t)( -csch(t)coth (x)/csch(t)]
-csch(t)coth (t) +csch(t)coth (x)ln(csch(t) + csch(t)coth (x)
f' = csch(t)coth (t)ln(csch(t)