Consider a triangle ABC such that `cotA+cotB+cotC=cot theta`. Now answer the following :
Q. `sin(A-theta)sin(B-theta) sin(C-theta)=`:

Answer

To solve the problem, we need to find the value of \( \sin(A - \theta) \sin(B - \theta) \sin(C - \theta) \) given that \( \cot A + \cot B + \cot C = \cot \theta \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We start with the equation \( \cot A + \cot B + \cot C = \cot \theta \). This can be rewritten using the identity for cotangent:
\[ \cot A = \frac{\cos A}{\sin A}, \quad \cot B = \frac{\cos B}{\sin B}, \quad \cot C = \frac{\cos C}{\sin C} \]
Thus,
\[ \frac{\cos A}{\sin A} + \frac{\cos B}{\sin B} + \frac{\cos C}{\sin C} = \frac{\cos \theta}{\sin \theta} \]

2. **Using the Sine Difference Identity**:
We apply the sine difference identity:
\[ \sin(A - \theta) = \sin A \cos \theta - \cos A \sin \theta \]
Similarly, we can write:
\[ \sin(B - \theta) = \sin B \cos \theta - \cos B \sin \theta \]
\[ \sin(C - \theta) = \sin C \cos \theta - \cos C \sin \theta \]

3. **Expressing the Product**:
Now we need to compute the product:
\[ \sin(A - \theta) \sin(B - \theta) \sin(C - \theta) \]
Substituting the expressions from the sine difference identities, we have:
\[ \sin(A - \theta) \sin(B - \theta) \sin(C - \theta) = (\sin A \cos \theta - \cos A \sin \theta)(\sin B \cos \theta - \cos B \sin \theta)(\sin C \cos \theta - \cos C \sin \theta) \]

4. **Simplifying the Expression**:
To simplify this product, we can denote:
\[ a_1 = \sin(A - \theta), \quad b_1 = \sin(B - \theta), \quad c_1 = \sin(C - \theta) \]
Thus, we can express:
\[ a_1 = \sin A \cos \theta - \cos A \sin \theta \]
\[ b_1 = \sin B \cos \theta - \cos B \sin \theta \]
\[ c_1 = \sin C \cos \theta - \cos C \sin \theta \]

5. **Finding the Product**:
The product \( a_1 b_1 c_1 \) can be expanded and simplified using the identities and the given condition. After simplification, we find that:
\[ \sin(A - \theta) \sin(B - \theta) \sin(C - \theta) = \sin^3 \theta \]

### Final Result:
Thus, the final value is:
\[ \sin(A - \theta) \sin(B - \theta) \sin(C - \theta) = \sin^3 \theta \]