Question

If $x+\frac{1}{x}=2\cos \theta$, then $x^3+\frac{1}{x^3}$ is equal to

• A. $\sin 3\theta$
• B. $2\sin 3\theta$
• C. $\cos 3\theta$
• D. $2\cos 3\theta$

Answer 1

The correct option is D $2\cos 3\theta$

Given, $x+\frac{1}{x}=2\cos \theta$
$\Rightarrow {(x+\frac{1}{x})}^3=(2\cos \theta {)}^3$
$\Rightarrow x^3+\frac{1}{x^3}+3x\times \frac{1}{x}(x+\frac{1}{x})=8{\cos }^2\theta$
$\Rightarrow x^3+\frac{1}{x^3}+3\times 2\cos \theta =8{\cos }^3\theta$
$\Rightarrow x^3+\frac{1}{x^3}=2(4{\cos }^2\theta -3\cos \theta )$
$=2\cos 3\theta$.