Question

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

$\mathbf{}\mathbf{tani}\mathbf{}\mathbf{z}\mathbf{=}\mathbf{i}\mathbf{}\mathbf{tanhz}$

Short answer

The equation$\mathrm{tani}\mathrm{z}=\mathrm{i}\mathrm{tanhz}$ is verified using the equations (11.4), (12.2) and (12.3).

Step-by-step solution

Given Information

Givenequationis$\mathrm{tani}\mathrm{z}=\mathrm{i}\mathrm{tanhz}$.

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.

 Simplify and Prove the Right Hand Side (RHS) of given equation.

Given the equation is,$\mathrm{tani}\mathrm{z}=\mathrm{i}\mathrm{tanhz}$ .

Write $\tan \mathrm{z}$as $\tan (z)=\frac{\sin (z)}{\cos (z)}$

Let $z\to zi$than$\tan (zi)=\frac{\sin (zi)}{\cos (zi)}\dots .\left(1\right)$ Use the following results.

$$\begin{gathered} \cos (iz)=\cos h(z) \\ \sin (iz)=i\sin h(z) \end{gathered}$$

Substitute the results in equation (1).

$$\begin{gathered} \tan (iz)=\frac{i\sin h(z)}{\cos h(z)} \\ \tan (iz)=i\tan h(z) \end{gathered}$$

Hence, the equation is verified.