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.

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

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

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

Question: Prove that cosh²x-sinh²x =1.

Answer:

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.

= e^x × e^-x

= e^x-x

= e⁰

= 1 = R.H.S.

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

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Formula of 1-cos2x	Formula of 1-sin2x
cos3x Formula	sin3x Formula
cos4x+sin4x Formula	1+tan2x Formula

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