define that or calculate it Cosh*2z-sinh*2z?. It`s answer should be cleared to make me understand.

We need to simplify the given expression:

cosh(2z) - sinh(2z)

Step 1: Recall Hyperbolic Identities
We use the standard hyperbolic function definitions:

Hyperbolic cosine:
cosh(x) = (e^x + e^(-x)) / 2

Hyperbolic sine:
sinh(x) = (e^x - e^(-x)) / 2

We also use the double-angle formulas:

cosh(2z) = cosh^2(z) + sinh^2(z)
sinh(2z) = 2 sinh(z) cosh(z)
Step 2: Substitute the Double Angle Formulas
Now, we apply these to the given expression:

cosh(2z) - sinh(2z)
= (e^(2z) + e^(-2z)) / 2 - (e^(2z) - e^(-2z)) / 2

Step 3: Simplify the Expression
Rewriting:

= (e^(2z) + e^(-2z) - e^(2z) + e^(-2z)) / 2
= (2e^(-2z)) / 2
= e^(-2z)

Conclusion:
The simplified result is:

cosh(2z) - sinh(2z) = e^(-2z)