Question

If `x + 1/x` = 2 cos θ, then `x^n + 1/x^n` is equal to ______.

Options

• 2 sin n θ

• 2 cos n θ

• sin (2n θ)

• cos (2n θ)

Answer 1

If `x + 1/x` = 2 cos θ, then `x^n + 1/x^n` is equal to 2 cos n θ.

Explanation:

Given, `x + 1/x` = 2 cos θ

`x^2 + 1/x^2 = (x + 1/x)^2 - 2` = 4 cos²θ – 2

`\implies x^2 + 1/x^2` = 2 cos 2 θ

Again, `x^3 + 1/x^3 + 3(x + 1/x)` = 8 cos³ θ

`\implies x^3 + 1/x^3` = 8 cos³ θ – 6 cos θ

= 2 cos 3 θ

Similalry, `x^n + 1/x^n` = 2 cos n θ