# Show that

Question:

Show that $(1-i)^{n}\left(1-\frac{1}{i}\right)^{n}=2^{n}$ for all n N.

Solution:

To show: $(1-i)^{n}\left(1-\frac{1}{i}\right)^{n}=2^{n}$

Taking LHS,

$(1-i)^{n}\left(1-\frac{1}{i}\right)^{n}$

$=(1-i)^{n}\left(1-\frac{1}{i} \times \frac{i}{i}\right)^{n} \quad$ [rationalize]

$=(1-i)^{n}\left(1-\frac{i}{i^{2}}\right)^{n}$

$=(1-i)^{n}\left(1-\frac{i}{-1}\right)^{n}\left[\because i^{2}=-1\right]$

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

$=[(1-i)(1+i)]^{n}$

$=\left[(1)^{2}-(i)^{2}\right]^{n}\left[(a+b)(a-b)=a^{2}-b^{2}\right]$

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

$=[1-(-1)]^{n}\left[\because i^{2}=-1\right]$

$=(2)^{n}$

$=2^{n}$

$=$ RHS

Hence Proved
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