Trigonometric functions are similar to Hyperbolic functions.  Hyperbolic functions also can be seen in many linear differential equations, for example in the cubic equations, the calculation of angles and distances in hyperbolic geometry are done through this formula. This has importance in electromagnetic theory, heat transfer, and special relativity.

The basic hyperbolic formulas are sinh, cosh, tanh.

$$e^x=coshx+sinhx$$

$$sinhx=\frac{e^x-e^{-x}}{2}$$

$$coshx=\frac{e^x+e^{-x}}{2}$$

$$tanhx=\frac{sinhx}{coshx}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$

 

RELATIONSHIPS AMONG HYPERBOLIC FUNCTION

Following is the relationship among hyperbolic function :

$$tanhx=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$

$$cothx=\frac{1}{tanhx}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$$

$$sechx=\frac{1}{coshx}=\frac{2}{e^x+e^{-x}}$$

$$cschx=\frac{1}{sinhx}=\frac{2}{e^x-e^{-x}}$$

 

Solved Examples

Question: Derive addition identities for sinh (x + y) and cos h(x + y)

In the identity: 

Solution:

Since,

Dividing numerator and denominator by