JEE 2024 — Mathematics PYQ

\begin{aligned}
 & \mathrm{-}z=2-i\left(2\tan{\frac{5\pi}{8}}\right) \\
 & =2-2i\frac{\sin\frac{5\pi}{8}}{\cos\frac{5\pi}{8}} \\
 & =\frac{2}{\cos\frac{5\pi}{8}}\left(\cos\frac{5\pi}{8}-i\sin\frac{5\pi}{8}\right) \\
 & & =\frac{2}{\cos\left(\pi-\frac{3\pi}{8}\right)}\left(\cos\left(\pi-\frac{3\pi}{8}\right)-i\sin\left(\pi-\frac{3\pi}{8}\right)\right) \\
 & =\frac{-2}{\mathrm{cos}\frac{3\pi}{8}}\left(-\mathrm{cos}\frac{3\pi}{8}-i\sin\frac{3\pi}{8}\right)
\end{aligned}