Show that the complex number z,

According to the question,

Let z = x + iy

arg ((z-1)/(z+1)) = π/4

⇒ arg (z – 1) – arg (z + 1) = π/4

⇒ arg (x + iy – 1) – arg (x + iy + 1) = π/4

⇒ arg (x – 1 + iy) – arg (x + 1 + iy) = π/4

$\Rightarrow \tan ^{-1} \frac{y}{x-1}+\tan ^{-1} \frac{y}{x+1}=\frac{\pi}{4}$

$\Rightarrow \tan ^{-1}\left[\frac{\frac{y}{x-1}-\frac{y}{x+1}}{1+\left(\frac{y}{x-1}\right)\left(\frac{y}{x+1}\right)}\right]=\frac{\pi}{4}$

$\Rightarrow \frac{y(x+1-x+1)}{x^{2}-1+y^{2}}=\tan \frac{\pi}{4}$

$\Rightarrow \frac{2 y}{x^{2}+y^{2}-1}=1$

⇒ x2 + y2 – 1 = 2y

⇒ x2 + y2 – 2y – 1 = 0

The equation obtained represents the equation of a circle.