Question

If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:
1. 2 cos θ
2. con nθ
3. 2 cos nθ
4. 2 sin nθ

Answer 1

Correct Answer - Option 3 : 2 cos nθ

Calculation:

Given x= cosθ + i sinθ

then xⁿ = (cos θ + i sinθ)ⁿ

⇒ xn = (cos nθ + i sin nθ)   -----(a)

⇒ \(\frac{1}{x^n}=\frac{1}{(cos nθ + i sin nθ)^1}= (cos nθ + i sin nθ)^-1= cos nθ - i sin nθ\)     -----(b)

on adding eqⁿ (a) and (b) we get,

⇒ \(x^n+\frac{1}{x^n}=\) cos nθ + i sin nθ + cos nθ -i sin nθ = 2 con nθ