`((sqrt(3)+2cosA)/(1-2sinA))^(-3)+((1+2sinA)/(sqrt(3)-2cosA))^(-3)=`____

Options

A. 1
B. `sqrt(3)`
C. `0`
D. `-1`

Answer

To solve the given expression

\[ \left(\frac{\sqrt{3} + 2\cos A}{1 - 2\sin A}\right)^{-3} + \left(\frac{1 + 2\sin A}{\sqrt{3} - 2\cos A}\right)^{-3}, \]

we will evaluate it step by step.

### Step 1: Substitute \( A = 90^\circ \)

First, we substitute \( A = 90^\circ \) into both fractions.

- For \( \cos 90^\circ = 0 \) and \( \sin 90^\circ = 1 \):

\[ \frac{\sqrt{3} + 2\cos 90^\circ}{1 - 2\sin 90^\circ} = \frac{\sqrt{3} + 2 \cdot 0}{1 - 2 \cdot 1} = \frac{\sqrt{3}}{1 - 2} = \frac{\sqrt{3}}{-1} = -\sqrt{3}. \]

- For the second fraction:

\[ \frac{1 + 2\sin 90^\circ}{\sqrt{3} - 2\cos 90^\circ} = \frac{1 + 2 \cdot 1}{\sqrt{3} - 2 \cdot 0} = \frac{1 + 2}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}. \]

### Step 2: Evaluate the powers

Now we can substitute these values back into the original expression:

\[ \left(-\sqrt{3}\right)^{-3} + \left(\sqrt{3}\right)^{-3}. \]

Calculating each term:

1. \((- \sqrt{3})^{-3} = -\frac{1}{(\sqrt{3})^3} = -\frac{1}{3\sqrt{3}}\).
2. \((\sqrt{3})^{-3} = \frac{1}{(\sqrt{3})^3} = \frac{1}{3\sqrt{3}}\).

### Step 3: Combine the results

Now, we combine the two results:

\[ -\frac{1}{3\sqrt{3}} + \frac{1}{3\sqrt{3}} = 0. \]

### Final Answer

Thus, the final result is:

\[ \boxed{0}. \]

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