Question

If $x+\frac{1}{x}=2\cos \theta$ then $x^6-\frac{1}{x^6}$ is

• A. $2i\sin 6\theta$
• B. $\cos 6\theta$
• C. $2i{\sin }^6\theta$
• D. $2{\cos }^6\theta$

Answer 1

The correct option is A $2i\sin 6\theta$

$x+\frac{1}{x}=2\cos \theta =e^{\theta }+e^{-i\theta }=e^{i\theta }\frac{1}{e^{-i\theta }}$

On comparing, we get $x=e^{i\theta }$

So, $x^6-\frac{1}{x^6}=e^{6i\theta }\frac{1}{e^{-6i\theta }}$$=(\cos 6\theta +i\sin 6\theta )$$-(\cos 6\theta -i\sin 6\theta )$

$⟹x^6-\frac{1}{x^6}=2i\sin 6\theta$