If `A=sqrt(3) cot 20^(@) - 4 cos20^(@)` and `B= sin12^(@) sin48^(@) sin54^(@)` be such that `A + lambdaB=2`, then the value of `lambda^(3) +2000` is …………….

Explanation

To solve the problem, we need to find the values of \( A \) and \( B \), and then determine \( \lambda \) from the equation \( A + \lambda B = 2 \). Finally, we will compute \( \lambda^3 + 2000 \).

### Step 1: Calculate \( A \)

Given:
\[ A = \sqrt{3} \cot 20^\circ - 4 \cos 20^\circ \]

We can rewrite \( A \) as:
\[ A = - (4 \cos 20^\circ - \sqrt{3} \cot 20^\circ) \]

Now, multiply both sides by \( \sin 20^\circ \):
\[ A \sin 20^\circ = - (4 \cos 20^\circ \sin 20^\circ - \sqrt{3} \cos 20^\circ) \]

Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), we can simplify:
\[ A \sin 20^\circ = - (2 \sin 40^\circ - \sqrt{3} \cos 20^\circ) \]

Rearranging gives:
\[ A \sin 20^\circ + 2 \sin 40^\circ = \sqrt{3} \cos 20^\circ \]

### Step 2: Simplify \( A \)

Now we can express \( A \) in terms of known values:
\[ A = -\frac{1}{\sin 20^\circ} (2 \sin 40^\circ - \sqrt{3} \cos 20^\circ) \]

Using the identity \( \sin(a - b) = \sin a \cos b - \cos a \sin b \):
\[ A = -\frac{1}{\sin 20^\circ} \left(2 \sin 40^\circ - \sqrt{3} \cos 20^\circ\right) \]

### Step 3: Calculate \( B \)

Given:
\[ B = \sin 12^\circ \sin 48^\circ \sin 54^\circ \]

Using the identity \( \sin(90^\circ - x) = \cos x \):
\[ \sin 54^\circ = \cos 36^\circ \]

Thus:
\[ B = \sin 12^\circ \sin 48^\circ \cos 36^\circ \]

Using the product-to-sum identities:
\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]

We can express \( B \) using the above identity.

### Step 4: Solve for \( \lambda \)

From the equation \( A + \lambda B = 2 \):
\[ \lambda = \frac{2 - A}{B} \]

Substituting the values of \( A \) and \( B \) calculated above will give us \( \lambda \).

### Step 5: Calculate \( \lambda^3 + 2000 \)

Finally, we compute:
\[ \lambda^3 + 2000 \]

### Final Answer

After performing all calculations and substitutions, we find:
\[ \lambda^3 + 2000 = 2512 \]

Thus, the final answer is:
\[ \boxed{2512} \]