Transformation between Hyperbolic and Trignometric function

* For hyperbolic sine:

Given that 

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

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

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

* For hyperbolic cosine:

Given that 

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

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

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

* For hyperbolic tangent:

Given that 

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

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

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

* For hyperbolic cotangent:

Given that 

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

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

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

* For hyperbolic secant:

Given that 

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

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

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

* For hyperbolic cosecant:

Given that 

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

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

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

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