Prove that Cosh(2x)=2cos2x-1?

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Answer 1 · 2018-03-06T10:35:57

See the proof below

I am going to prove the following

$cosh2x=2cosh^2x-1$

By the definition of hyperbolic functions

$cosh(2x)=(e^(2x)+e^(-2x))/2$

$coshx=(e^x+e^-x)/(2)$

$cosh^2x=((e^x+e^-x)/(2))^2$

$=1/4(e^x+e^-x)^2$

$=1/4(e^(2x)+e^(-2x)+2)$

Therefore,

$2cosh^2x-1=2*1/4(e^(2x)+e^(-2x)+2)-1$

$=(e^(2x)+e^(-2x))/2+1-1$

$=cosh(2x)$

$QED$