Question

if[1-i/1+i]^100=(a+ib) , find the values of a and b

Answer 1

Rationalising [(1 - i)/(1 + i)],
$\frac{1 \: - \: i}{1 \: + \: i} \: \times \: \frac{1 \: - \: i}{1 \: - \: i}$
$\frac{{(1 \: - \: i)}^{2} }{{(1)}^{2} \: + \: {( - 1)}^{2} }$
$\frac{(1 \: - \: 1 \: - \: 2i)}{2}$
→ – i
${( - \: i)}^{100}$
${( - \: i)}^{4 \: \times \: 25}$
Since (– i)⁴ = (i)⁴ = 1,
Therefore,
${( - \: i)}^{100} \: = \: 1$
→ 1
→ (1) + (0)i = a + ib
Therefore,
a = 1, b = 0.