If 1, + ,sin x, + ,{sin ^2}x, + ,...∞ , = ,4, + ,2√ 3 ,,0, < ,x, < ,π then

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8. Sequences and Series

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

A

$x = \frac{\pi }{6}$

B

$x = \frac{\pi }{3}$

C

$x = \frac{\pi }{6}$ or $\frac{\pi }{3}$

D

none of these

Solution

$1+\sin x+\sin ^{2} x \ldots \ldots \ldots=4+2 \sqrt{3}$

$\Rightarrow \frac{1}{1-\sin x}=4+2 \sqrt{3}$

$\Rightarrow \sin x=-\frac{\sqrt{3}}{2}$

$\Rightarrow$ No value of $x$ in $(0, \pi)$

Standard 11

Mathematics

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