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

Before we prove the identity cosh^{2}x-sinh^{2}x=1, let us recall that

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

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

**Question:** Prove that cosh^{2}x-sinh^{2}x =1.

**Answer:**

By the above two formulas, we have that

L.H.S = cosh^{2}x – sinh^{2}x

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

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

= 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}

= e^{0}

= 1 = R.H.S.

Hence, we have the proved the identity cosh^{2}x-sinh^{2}x =1.

**Related Formulas:**

Formula of 1-cos^{2}x | Formula of 1-sin^{2}x |

cos3x Formula | sin3x Formula |

cos^{4}x+sin^{4}x Formula | 1+tan^{2}x Formula |

## FAQs

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

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