If \( 1+2 \sin x+3 \sin ^{2} x+4 \sin ^{3} x+\ldots \) upto infinite terms \( =4 \) and number o...

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If \( 1+2 \sin x+3 \sin ^{2} x+4 \sin ^{3} x+\ldots \) upto infinite terms \( =4 \) and number of solutions of the equation in \( \left[\frac{-3 \pi}{2}, 4 \pi\right] \) is \( k \).
The value of \( \left|\frac{\cos 2 x-1}{\sin 2 x}\right| \) is equal to
(a) 1
(b) \( \sqrt{3} \)
(c) \( 2-\sqrt{3} \)
(d) \( \frac{1}{\sqrt{3}} \)
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