How can you prove #sinh(2x)coth(x) = cosh(2x)+1#?

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Answer 1 · 2018-04-21T14:24:17

Use identities and axioms to change only one side until it is identical to the other side.

Prove:

#sinh(2x)coth(x) = cosh(2x)+1#

Substitute #sinh(2x) = 2sinh(x)cosh(x)#

#2sinh(x)cosh(x)coth(x) = cosh(2x)+1#

Substitute #coth(x) = cosh(x)/sinh(x)#

#2sinh(x)cosh(x)cosh(x)/sinh(x) = cosh(2x)+1#

#sinh(x)/sinh(x) to 1#:

#2cosh(x)cosh(x) = cosh(2x)+1#

Write as a square:

#2cosh^2(x) = cosh(2x)+1#

One identity for #cosh(2x)# is #cosh(2x) = 2cosh^2(x)-1#, therefore, we may substitute #2cos^2(x) = cosh(2x) +1#:

#cosh(2x)+1 = cosh(2x)+1# Q.E.D.