How to Prove cosh^2x-sinh^2x=1

The formula of cosh2x-sinh2x is given by cosh2x-sinh2x =1. Here we will learn how to prove cosh^2x-sinh^2x=1.

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Formula of cosh^2x-sinh^2x

Before we prove the identity cosh2x-sinh2x=1, let us recall that

  1. coshx = (ex +e-x)/2
  2. sinhx = (ex – e-x)/2.

Proof of cosh^2x-sinh^2x=1

Question: Prove that cosh2x-sinh2x =1.


By the above two formulas, we have that

L.H.S = cosh2x – sinh2x

= $\Big(\dfrac{e^x+e^{-x}}{2} \Big)^2$ $- \Big(\dfrac{e^x-e^{-x}}{2} \Big)^2$

= $\Big(\dfrac{e^x}{2}+\dfrac{e^{-x}}{2} \Big)^2$ $- \Big(\dfrac{e^x}{2}-\dfrac{e^{-x}}{2} \Big)^2$

= 4 × $\dfrac{e^x}{2}$ × $\dfrac{e^{-x}}{2}$ using the formula (a+b)2 – (a-b)2 =4ab.

= ex × e-x

= ex-x

= e0

= 1 = R.H.S.

Hence, we have the proved the identity cosh2x-sinh2x =1.

Related Formulas:

Formula of 1-cos2xFormula of 1-sin2x
cos3x Formulasin3x Formula
cos4x+sin4x Formula1+tan2x Formula


Q1: What is the formula of cosh^2x-sinh^2x?

Answer: The formula/identity of cosh^2x-sinh^2x is given by cosh2x-sinh2x =1.

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