Question

show that
(1-i)^n (1 - 1/i)^n = 2^n
for all nEN​

Answer 1

Step-by-step explanation:

LHS

$(1 - i) ^n(1 - \frac{1}{i} ) {}^{n} \\ ={((1 - i)(1 - \frac{1}{i} ) } ){}^{n} \\ = (1 - \frac{1}{i} - i + \frac{i}{i} ) ^{n} \\ = (2 - ( \frac{1}{i} + i)) ^{n} \\ = (2 - ( \frac{1 + i {}^{2} }{i} )) {}^{n} \\ = (2 -( \frac{1 - 1}{i})) {}^{n} \\ =(2 - 0) {}^{n} \\ = 2 {}^{n}$