Question

(1+i) (1-i)-¹
express the following in the form of a+ib,a, beR i=√-1 ​

Answer 1

Step-by-step explanation:

We have,

$z = (1 + i)(1 - i) ^{ - 1}$

$\implies \: z = \sqrt{2} ( \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{2} } i). \sqrt{2} ( \frac{1}{ \sqrt{2} } - \frac{1}{ \sqrt{2} } i)^{ - 1 }$

$\implies \: z = 2( \cos( \frac{\pi}{4} ) + \sin( \frac{\pi}{4} ) i)( \cos( - \frac{\pi}{4} ) + \sin( - \frac{\pi}{4} ) )^{ - 1}$

$\implies \: z = 2 .{e}^{ \frac{i\pi}{4} } . ({e}^{ \frac{ -i\pi}{4} } )^{ - 1}$

$\implies \: z = 2. {e}^{ \frac{i\pi}{4} } . {e}^{ \frac{i\pi}{4} }$

$\implies \: z = 2 {e}^{ \frac{i\pi}{2} }$

$\implies \: z = 2( \cos( \frac{\pi}{2} ) + i \sin( \frac{\pi}{2} ) )$

$\implies \: z = 2(0 + i)$

$\implies \: z = 2i$