` sum_(k=0)^5(1)/(sin[(K+1)theta]sin[(K+2)theta])` can be equal to

Options

A. `(cot2theta-cot7theta)/(sin theta)`
B. `(tan2theta-tan7theta)/(sin2theta)`
C. a positive number
D. a negative number

Answer

To solve the problem \( \sum_{k=0}^{5} \frac{1}{\sin((k+1)\theta) \sin((k+2)\theta)} \), we can use a trigonometric identity to simplify the expression.

### Step-by-Step Solution:

1. **Write the Summation**:
\[ S = \sum_{k=0}^{5} \frac{1}{\sin((k+1)\theta) \sin((k+2)\theta)} \]

2. **Use the Identity**:
We can use the identity:
\[ \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \]
However, in this case, we can also use the identity:
\[ \frac{1}{\sin A \sin B} = \frac{1}{\sin A} \cdot \frac{1}{\sin B} \]

3. **Rewrite the Terms**:
Each term in the summation can be rewritten using the cotangent function:
\[ \frac{1}{\sin((k+1)\theta) \sin((k+2)\theta)} = \frac{1}{\sin((k+1)\theta)} \cdot \frac{1}{\sin((k+2)\theta)} \]

4. **Apply the Sum**:
We can express the sum as:
\[ S = \sum_{k=0}^{5} \left( \cot((k+1)\theta) - \cot((k+2)\theta) \right) \]

5. **Evaluate the Telescoping Series**:
Notice that this is a telescoping series:
\[ S = \left( \cot(\theta) - \cot(7\theta) \right) \]

6. **Final Expression**:
Thus, the final result of the sum is:
\[ S = \cot(\theta) - \cot(7\theta) \]

### Conclusion:
The sum \( \sum_{k=0}^{5} \frac{1}{\sin((k+1)\theta) \sin((k+2)\theta)} \) simplifies to \( \cot(\theta) - \cot(7\theta) \).