Prove: ##(\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)##(adsbygoogle = window.adsbygoogle || []).push({});

Newton's binomial: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## and: ##(a-b)^n~\rightarrow~(-1)^kC^k_n##

I ignore the coefficients.

$$(\cosh(x)+\sinh(x))^n=\cosh^n(x)+\cosh^{n-1}\sinh(x)+...+\sinh^n(x)$$

$$\cosh^n(x)=(e^x+e^{-x})^n=e^{nx}+e^{(n-2)x}+e^{(n-4)x}+...+e^{-nx}$$

$$\cosh^{n-1}\sinh(x)=(e^x+e^{-x})^{n-1}(e^x-e^{-x})=...=e^{nx}-e^{-nx}$$

I use this result in the next derivations:

$$\cosh^{(n-2)}x\sinh^2(x)=[(e^x+e^{-x})^{n-1}(e^x-e^{-x})](e^x-e^{-x})=...=e^{nx}+e^{-nx}-e^{(n-2)x}-e^{-(n-2)x}$$

$$\cosh^{(n-3)}x\sinh^3(x)=[(e^x+e^{-x})^{n-1}(e^x-e^{-x})](e^x-e^{-x})^2=...=e^{nx}-e^{-nx}+2e^{-(n-2)x}+e^{(n-4)x}-2e^{(n-2)x}-e^{-(n-4)x}$$

It doesn't lead anywhere

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I An identity of hyperbolic functions

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - identity hyperbolic functions | Date |
---|---|

I Intuitive understanding of Euler's identity? | Thursday at 4:52 AM |

How to prove vector identities WITHOUT using levi civita ? | Nov 27, 2017 |

B Hot tubs and hyperbolic curves | Nov 25, 2017 |

I Domain of the identity function after inverse composition | Nov 19, 2017 |

B Algebra Identity I can't figure out | Oct 29, 2017 |

**Physics Forums - The Fusion of Science and Community**