Finding the cosh2 z - sinh2 z value

The question asks for the value of the expression $\cosh^2 z - \sinh^2 z$. This expression involves hyperbolic functions, which are fundamental in various areas of mathematics, including calculus and the study of complex numbers.

To find the value, we can use the definitions of the hyperbolic cosine and hyperbolic sine functions in terms of exponential functions:

• The hyperbolic cosine of $z$ is defined as $\cosh z = \frac{e^z + e^{-z}}{2}$.
• The hyperbolic sine of $z$ is defined as $\sinh z = \frac{e^z - e^{-z}}{2}$.

Now, let's square both definitions:

First, square $\cosh z$:

$\cosh^2 z = \left(\frac{e^z + e^{-z}}{2}\right)^2 = \frac{(e^z)^2 + 2(e^z)(e^{-z}) + (e^{-z})^2}{4} = \frac{e^{2z} + 2e^{z-z} + e^{-2z}}{4} = \frac{e^{2z} + 2e^0 + e^{-2z}}{4} = \frac{e^{2z} + 2(1) + e^{-2z}}{4} = \frac{e^{2z} + 2 + e^{-2z}}{4}$.

Next, square $\sinh z$:

$\sinh^2 z = \left(\frac{e^z - e^{-z}}{2}\right)^2 = \frac{(e^z)^2 - 2(e^z)(e^{-z}) + (e^{-z})^2}{4} = \frac{e^{2z} - 2e^{z-z} + e^{-2z}}{4} = \frac{e^{2z} - 2e^0 + e^{-2z}}{4} = \frac{e^{2z} - 2(1) + e^{-2z}}{4} = \frac{e^{2z} - 2 + e^{-2z}}{4}$.

Now, we subtract $\sinh^2 z$ from $\cosh^2 z$ to find the cosh2 z - sinh2 z value:

$\cosh^2 z - \sinh^2 z = \frac{e^{2z} + 2 + e^{-2z}}{4} - \frac{e^{2z} - 2 + e^{-2z}}{4}$

Combine the terms over a common denominator:

$\cosh^2 z - \sinh^2 z = \frac{(e^{2z} + 2 + e^{-2z}) - (e^{2z} - 2 + e^{-2z})}{4}$

Distribute the negative sign in the numerator:

$\cosh^2 z - \sinh^2 z = \frac{e^{2z} + 2 + e^{-2z} - e^{2z} + 2 - e^{-2z}}{4}$

Cancel out the $e^{2z}$ and $e^{-2z}$ terms:

$\cosh^2 z - \sinh^2 z = \frac{2 + 2}{4} = \frac{4}{4} = 1$.

Understanding the Hyperbolic Identity: cosh2 z - sinh2 z value

The result $\cosh^2 z - \sinh^2 z = 1$ is a fundamental identity in hyperbolic trigonometry. This identity is analogous to the Pythagorean identity in standard trigonometry, $\cos^2 \theta + \sin^2 \theta = 1$, but with a minus sign due to the definitions of the hyperbolic functions.

This identity holds true for any value of $z$, whether it is a real number or a complex number. Its derivation relies purely on the exponential definitions of $\cosh z$ and $\sinh z$, concepts covered in calculus.

Thus, the cosh2 z - sinh2 z value is consistently 1.

Comparing this result to the given options, the correct value is 1.