If \({\rm X} + \frac{1}{{\rm X}}\) = 2 cos \(\theta \), then \({{\rm X}^3} + \frac{1}{{{{\rm X}^3}}}\) = ?

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 03 Dec 2022 Shift 2)

Answer 1

Given:

\({\rm X} + \frac{1}{{\rm X}}\) = 2 cos \(θ \)

Concept used:

4cos3θ - 3cosθ = cos3θ

\({{\rm X}^3} + \frac{1}{{{{\rm X}^3}}}\) = \(\left({\rm X} + \frac{1}{{\rm X}}\right)^3 \) - 3\(\left({\rm X} + \frac{1}{{\rm X}}\right) \)

Calculation:

\({{\rm X}^3} + \frac{1}{{{{\rm X}^3}}}\) = (2 cosθ)³ - 3(2cosθ)

⇒ 8cos3θ - 6cosθ

⇒ 2(4cos3θ - 3cosθ)

⇒ 2cos3θ

∴ The required answer is 2 cos3θ.

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