Question

If $x+\frac{1}{x}=2\cos \theta ,$ then find the value of $x^3+\frac{1}{x^3}.$

• A. $cos3\theta$
• B. $2cos3\theta$
• C. $\frac{1}{2}cos3\theta$
• D. $\frac{1}{3}cos3\theta$

Answer 1

The correct option is B $2cos3\theta$

We have $x+\frac{1}{x}=2\cos \theta ,$
Now, using the identity:
$a^3+b^3=(a+b{)}^3-3ab(a+b)$

$\Rightarrow x^3+\frac{1}{x^3}={(x+\frac{1}{x})}^3-3x\times \frac{1}{x}(x+\frac{1}{x})$

$\Rightarrow x^3+\frac{1}{x^3}=(2\cos \theta {)}^3-3\times 2\cos \theta$

$\Rightarrow x^3+\frac{1}{x^3}=8{\cos }^3\theta -6\cos \theta$
$\Rightarrow x^3+\frac{1}{x^3}=2(4{\cos }^3\theta -3\cos \theta )$
$\Rightarrow x^3+\frac{1}{x^3}=2\cos 3\theta$