If the equation `sin6x + cos4x=-2` have a family of nonnegative solutions `x_(k)'s`, where `0 le x_(1) le x_(2) le x_(3)`……………… `le x_(k) le x_(k+1)…………….` , then the value of `1/pi sum_(k=1)^(1000) x_(k+1) - x_(k)|` is…………..

Explanation

To solve the equation \( \sin 6x + \cos 4x = -2 \) and find the value of \( \frac{1}{\pi} \sum_{k=1}^{1000} (x_{k+1} - x_k) \), we will follow these steps:

### Step 1: Analyze the Equation
The equation \( \sin 6x + \cos 4x = -2 \) can only be satisfied if both \( \sin 6x \) and \( \cos 4x \) take their minimum values, which are -1. Therefore, we need:

\[ \sin 6x = -1 \quad \text{and} \quad \cos 4x = -1 \]
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