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

sin 2z=2 sin z cos z

The given equation is sin 2z=2 sin z cos z.

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.

The exponential form of the given equation is,

$\sin 2z=\frac{e^{2zi}-e^{-2zi}}{2i}$ …. (1)

Use exponential rule toexpand the equation (1).

$\sin 2z=\frac{e^{zi}.e^{zi}-e^{-zi}.e^{-zi}}{2i}$ …. (2)

Add $\left(e^{zi}.e^{-zi}-e^{zi}.e^{-zi}\right)$to the numerator of equation (2).

$$\begin{gathered} \sin 2z=\frac{\left[e^{zi}.e^{zi}\right]-\left[e^{-zi}.e^{-zi}\right]+\left[e^{zi}.e^{-zi}\right]-\left[e^{zi}.e^{-zi}\right]}{2i} \\ \sin 2z=\frac{\left[e^{zi}.e^{zi}\right]-\left[e^{-zi}.e^{-zi}\right]+\left[e^{zi}.e^{-zi}\right]-\left[e^{zi}.e^{-zi}\right]}{2i} \end{gathered}$$

Take $e^{zi}$and $e^{-zi}$common.

$$\begin{gathered} \sin 2z=\frac{e^{zi}\left(e^{zi}-e^{-zi}\right)+e^{-zi}\left(e^{zi}+e^{-zi}\right)}{2i} \\ \sin 2z=\frac{\left(e^{zi}-e^{-zi}\right)\left(e^{zi}+e^{-zi}\right)}{2i} \end{gathered}$$

Multiply the above equation by $\frac{2}{2}$.

$\sin 2z=\frac{\left(e^{zi}-e^{-zi}\right)\left(e^{zi}+e^{-zi}\right)}{2i}\times \frac{2}{2}$ … (3)

Rearrange the equation (3).

$$\begin{gathered} \sin 2z=2\times \frac{\left(e^{zi}-e^{-zi}\right)}{2i}\times \frac{\left(e^{zi}+e^{-zi}\right)}{2} \\ \sin 2z=2\sin z\cos z \end{gathered}$$

Hence the equation is verified.