Question

Find the modulus, argument and the principal argument of the complex number.
$(\tan 1-i{)}^2$.

Answer 1

$(\tan 1-i{)}^2$

$={\tan }^{2}1-1-2i\tan 1$

Modulus $=\sqrt{(ta{n}^{2}1-1{)}^2+4{\tan }^{2}1}$

$=({\tan }^{2}1+1)$

Argument= ${\tan }^{-1}\left(\frac{-2\tan 1}{{\tan }^{2}1-1}\right)$

Arg $={\tan }^{-1}\left(n\left(\frac{\tan 1+\tan 1}{1-\tan 1\tan 1}\right)\right)$

$={\tan }^{-1}\left(\tan 2\right)$

$=2$ .