Question

If $1+\sin x+{\sin }^{2}x+{\sin }^{3}x+....\infty$ is equal to $4+2\sqrt{3},0<x<\pi$, then $x$ is equal to

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$ or $\frac{\pi }{6}$
• D. $\frac{\pi }{3}$ or $\frac{2\pi }{3}$

Answer 1

The correct option is D $\frac{\pi }{3}$ or $\frac{2\pi }{3}$

We have
$1+\sin x+{\sin }^{2}x+{\sin }^{3}x+...=4+2\sqrt{3}$
$\Rightarrow \frac{1}{1-\sin x}=4+2\sqrt{3}$
$\Rightarrow \sin x=1-\frac{1}{4+2\sqrt{3}}=\frac{\sqrt{3}}{2}$
$\Rightarrow x=\frac{\pi }{3}$ or $\frac{2\pi }{3}$