Question : If $x+\frac{1}{x}=2 \cos \theta$, then $x^3+\frac{1}{x^3}=?$
Option 1: $2 \cos 2θ$
Option 2: $\cos 3θ$
Option 3: $2 \cos 3θ$
Option 4: $\cos 2θ$
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Correct Answer: $2 \cos 3θ$
Solution :
Given: $x+\frac{1}{x}=2 \cos \theta$.
Cubing both sides, we get:
$⇒\left(x^3+\frac{1}{x^3}\right) + 3\left(x+\frac{1}{x}\right) = 8\cos^3 \theta$
Putting the values, we get:
$⇒x^3+\frac{1}{x^3} = 8\cos^3 \theta- 3(2 \cos \theta)$
$⇒x^3+\frac{1}{x^3}= 8\cos^3 \theta- 6 \cos \theta$
$⇒x^3+\frac{1}{x^3}=2(4\cos^3 \theta- 3 \cos \theta)$
$⇒x^3+\frac{1}{x^3}=2 \cos 3 \theta$
Hence, the correct answer is $2 \cos 3 \theta$.
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