Question

If ${\left(\frac{1+i}{1-i}\right)}^x=1$ then

• A. $x=2n,n\in N$
• B. $x=4n+1,I,n\in N$
• C. $x=2n+1,n\in N$
• D. $x=4n,n\in N$

Answer 1

The correct option is D $x=4n,n\in N$

Using Euler's form, we get
${\left(\frac{e^{i\frac{\pi }{4}}}{e^{-i\frac{\pi }{4}}}\right)}^x=e^{i2n\pi }$
Where, $n$ is a natural number.
${\left(e^{i\frac{\pi }{2}}\right)}^x=e^{i2n\pi }$
$e^{ix\frac{\pi }{2}}=e^{i2n\pi }$
Therefore,
$x\frac{\pi }{2}=2n\pi$
$x=4n$
Where $nϵN$.
Hence, option 'D' is correct.