Let r and θ respectively be the modulus and amplitude of the complex number z=2-i(2 tan 5π 8), then (r,θ ) is equal to

Given: $z=2-i(2\mathrm{tan}\frac{5\pi }{8})$z=2−i(2tan85π​)

$$\Rightarrow z=2[1-i\mathrm{tan}(\frac{5\pi }{8})]$$⇒z=2[1−itan(85π​)]

$$\Rightarrow z=2[1-\frac{i\mathrm{sin}(\frac{5\pi }{8})}{\mathrm{cos}(\frac{5\pi }{8})}]$$⇒z=2[1−cos(85π​)isin(85π​)​]

$$\Rightarrow z=\frac{2}{\mathrm{cos}(\frac{5\pi }{8})}[\mathrm{cos}(\frac{5\pi }{8})-i\mathrm{sin}(\frac{5\pi }{8})]$$⇒z=cos(85π​)2​[cos(85π​)−isin(85π​)]

$$\Rightarrow z=\frac{2}{\mathrm{cos}(\pi -\frac{3\pi }{8})}[\mathrm{cos}(\pi -\frac{3\pi }{8})-i\mathrm{sin}(\pi -\frac{3\pi }{8})]$$⇒z=cos(π−83π​)2​[cos(π−83π​)−isin(π−83π​)]

$$\Rightarrow z=\frac{2}{-\mathrm{cos}(\frac{3\pi }{8})}[-\mathrm{cos}(\frac{3\pi }{8})-i\mathrm{sin}(\frac{3\pi }{8})]$$⇒z=−cos(83π​)2​[−cos(83π​)−isin(83π​)]

$$\Rightarrow z=\frac{2}{\mathrm{cos}(\frac{3\pi }{8})}[\mathrm{cos}(\frac{3\pi }{8})+i\mathrm{sin}(\frac{3\pi }{8})]$$⇒z=cos(83π​)2​[cos(83π​)+isin(83π​)]

$$\Rightarrow z=2\mathrm{sec}\frac{3\pi }{8}{e}^{i\cdot \frac{3\pi }{8}}$$⇒z=2sec83π​ei⋅83π​

Now on comparing with $z=|z|{e}^{i\theta }$z=∣z∣eiθ we get,

$$\Rightarrow \theta =\frac{3\pi }{8},r=2\mathrm{sec}\frac{3\pi }{8}$$⇒θ=83π​,r=2sec83π​