Prove that (cosh x + sinh x)ⁿ = cosh (nx) + sinh (nx)

by RoRi

November 18, 2015

Prove the following identity:

$(\cosh x + \sinh x)^n = \cosh (nx) + \sinh (nx).$

Proof. We know from a previous exercise (Section 6.19, Exercise #9) that

$\cosh x + \sinh x = e^x \quad \implies \quad \cosh (nx) + \sinh(nx) = e^{nx}.$

Therefore, we have

$(\cosh x + \sinh x)^n = (e^x)^n = e^{nx} = \cosh (nx) + \sinh(nx). \qquad \blacksquare$

Stumbling Robot