$$\begin{aligned} & z=2-i\left(2 \tan \frac{5 \pi}{8}\right)=x+i y(\text { let }) \\ & r=\sqrt{x^2+y^2} ~\& ~\theta=\tan ^{-1} \frac{y}{x} \\ & r=\sqrt{(2)^2+\left(2 \tan \frac{5 \pi}{8}\right)^2} \\ & =\left|2 \sec \frac{5 \pi}{8}\right|=\left|2 \sec \left(\pi-\frac{3 \pi}{8}\right)\right| \\ & =2 \sec \frac{3 \pi}{8} \\ & \& ~\theta = {\tan ^{ - 1}}\left( {{{ - 2\tan {{5\pi } \over 8}} \over 2}} \right) \\ & =\tan ^{-1}\left(\tan ^2\left(\pi-\frac{5 \pi}{8}\right)\right) \\ & =\frac{3 \pi}{8} \end{aligned}$$