Q8P - Mathematical Methods in Physical Sciences

Question

Verify each of the following by using equations (11.4), (12.2), and (12.3).

 $\mathbf{cos}\mathit{h}\mathbf{ }\mathbf{2}\mathit{z}\mathbf{=}{\mathbf{cosh}}^{\mathbf{2}}\mathbf{ }\mathbf{z}\mathbf{+}{\mathbf{sinh}}^{\mathbf{2}}\mathbf{z}$

Step-by-Step Solution

Answer

The equation $\cosh 2\mathrm{z}={\cosh }^{2 }\mathrm{z}+{\sinh }^2 \mathrm{z}$ is verified using the equations (11.4), (12.2) and (12.3).

Step 1: Given information

The given function $\cosh 2\mathrm{z}={\cosh }^2\mathrm{z}+{\sinh }^2\mathrm{z} .$

Step 2: Definition of Hyperbolic Function.

A hyperbolic function is a representation of the relationship between a point's distances from the origin to the coordinate axes as a function of an angle.

Step 3: Use exponential form to expand the equation

The exponential form of the given equation is,

$\cos h 2z=\frac{e^{2z}+e^{-2z}}{2}$                                                                                                   …. (1)

Multiply the equation (1) by $\frac{2}{2}$.

$$\begin{gathered} \cosh 2\mathrm{z}=\frac{{\mathrm{e}}^{2\mathrm{z}}+{\mathrm{e}}^{-2\mathrm{z}}}{2}\times \frac{2}{2} \\ \cosh 2\mathrm{z}=\frac{2{\mathrm{e}}^{2\mathrm{z}}+2{\mathrm{e}}^{-2\mathrm{z}}}{4} \end{gathered}$$

                                                                                                  …(2)

Add$2\left[e^z{e}^{-z}\right]-2\left[e^z{e}^{-z}\right]$ to square the numerator of equation (2).

$$\begin{gathered} \cosh \mathrm{z}=\frac{2{\mathrm{e}}^{2\mathrm{z}}+2{\mathrm{e}}^{2\mathrm{z}}+2\left[{\mathrm{e}}^{\mathrm{z}}{\mathrm{e}}^{\mathrm{z}}\right]-2\left[{\mathrm{e}}^{\mathrm{z}}{\mathrm{e}}^{\mathrm{z}}\right]}{4} \\ \cosh \mathrm{z}=\frac{{\mathrm{e}}^{2\mathrm{z}}+{\mathrm{e}}^{2\mathrm{z}}+2\left[{\mathrm{e}}^{\mathrm{z}}{\mathrm{e}}^{\mathrm{z}}\right]-2\left[{\mathrm{e}}^{\mathrm{z}}{\mathrm{e}}^{\mathrm{z}}\right]}{4}+\frac{{\mathrm{e}}^{2\mathrm{z}}+{\mathrm{e}}^{2\mathrm{z}}-2\left[{\mathrm{e}}^{\mathrm{z}}{\mathrm{e}}^{\mathrm{z}}\right]}{4} \\ \cosh \mathrm{z}={\cosh }^2 \mathrm{z}+{\sinh }^2\mathrm{z} \end{gathered}$$

 

Hence the equation is verified.