Q16 of 202 Page 171

Show that for all n N.

To show:

Taking LHS,

[rationalize]

[∵ i² = -1]

= (1 – i)ⁿ(1 + i)ⁿ

= [(1 – i)(1 + i)]ⁿ

= [(1)² – (i)²]ⁿ [(a + b)(a – b) = a² – b²]

= (1 – i²)ⁿ

= [1 – (-1)]ⁿ[∵ i² = -1]

= (2)ⁿ

= 2ⁿ

= RHS

Hence Proved

If, prove that x² + y² = 1.

If , where c is real, prove that a² + b² = 1 and .

Find the smallest positive integer n for which (1 + i)²ⁿ = (1 – i)²ⁿ.

Prove that (x + 1 + i) (x + 1 – i) (x – 1 – i) (x – 1 – i) = (x⁴ + 4).