Transformation between Hyperbolic and Trignometric function

* For hyperbolic sine:

Given that 

$$\sinh x = -i \sin (i x)$$

We substitute $x$ with $ix$ to get:

$$\sinh (ix) = i \sin x$$

* For hyperbolic cosine:

Given that 

$$\cosh x = \cos (i x)$$

We substitute $x$ with $ix$ to get:

$$\cosh (ix) = \cos x$$

* For hyperbolic tangent:

Given that 

$$\tanh x = -i \tan (i x)$$

We substitute $x$ with $ix$ to get:

$$\tanh (ix) = i \tan x$$

* For hyperbolic cotangent:

Given that 

$$\coth x = i \cot (i x)$$

We substitute $x$ with $ix$ to get:

$$\coth (ix) = -i \cot x$$

* For hyperbolic secant:

Given that 

$$\operatorname{sech} x = \sec (i x)$$

We substitute $x$ with $ix$ to get:

$$\operatorname{sech} (ix) = \sec x$$

* For hyperbolic cosecant:

Given that 

$$\operatorname{csch} x = i \csc (i x)$$

We substitute $x$ with $ix$ to get:

$$\operatorname{csch} (ix) = -i \csc x$$

In each of the transformations, we have used the fact that $i^2 = -1$.