If `1+sinx+sin^2x+sin^3x+oo` is equal to `4+2sqrt(3),0

Options

A. `pi/6`
B. `pi/4`
C. `pi/3or pi/6`
D. `pi/3 or (2pi)/3`

Explanation

To solve the problem, we need to find the value of \( x \) such that the infinite series \( 1 + \sin x + \sin^2 x + \sin^3 x + \ldots \) equals \( 4 + 2\sqrt{3} \) for \( 0 < x < \pi \).

### Step-by-Step Solution:

1. **Identify the Series**: The series \( 1 + \sin x + \sin^2 x + \sin^3 x + \ldots \) is a geometric series with the first term \( a = 1 \) and the common ratio \( r = \sin x \).

2. **Sum of Infinite Geometric Series**: The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
where \( |r| < 1 \). Here, \( a = 1 \) and \( r = \sin x \), so:
\[ S = \frac{1}{1 - \sin x} \]

3. **Set Up the Equation**: We set the sum equal to \( 4 + 2\sqrt{3} \):
\[ \frac{1}{1 - \sin x} = 4 + 2\sqrt{3} \]

4. **Cross Multiply**: To eliminate the fraction, we cross multiply:
\[ 1 = (4 + 2\sqrt{3})(1 - \sin x) \]

5. **Expand the Right Side**: Distributing the right side gives:
\[ 1 = (4 + 2\sqrt{3}) - (4 + 2\sqrt{3})\sin x \]

6. **Rearrange the Equation**: Rearranging the equation to isolate \( \sin x \):
\[ (4 + 2\sqrt{3})\sin x = (4 + 2\sqrt{3}) - 1 \]
\[ (4 + 2\sqrt{3})\sin x = 3 + 2\sqrt{3} \]

7. **Solve for \( \sin x \)**: Divide both sides by \( 4 + 2\sqrt{3} \):
\[ \sin x = \frac{3 + 2\sqrt{3}}{4 + 2\sqrt{3}} \]

8. **Rationalize the Denominator**: To simplify, multiply the numerator and denominator by the conjugate of the denominator:
\[ \sin x = \frac{(3 + 2\sqrt{3})(4 - 2\sqrt{3})}{(4 + 2\sqrt{3})(4 - 2\sqrt{3})} \]

9. **Calculate the Denominator**: The denominator simplifies as follows:
\[ (4 + 2\sqrt{3})(4 - 2\sqrt{3}) = 16 - 12 = 4 \]

10. **Calculate the Numerator**: The numerator expands to:
\[ (3)(4) + (3)(-2\sqrt{3}) + (2\sqrt{3})(4) + (2\sqrt{3})(-2\sqrt{3}) = 12 - 6\sqrt{3} + 8\sqrt{3} - 12 = 2\sqrt{3} \]

11. **Final Expression for \( \sin x \)**:
\[ \sin x = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \]

12. **Find \( x \)**: The values of \( x \) for which \( \sin x = \frac{\sqrt{3}}{2} \) in the interval \( 0 < x < \pi \) are:
\[ x = \frac{\pi}{3} \]

### Conclusion:
Thus, the value of \( x \) is \( \frac{\pi}{3} \).