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 \( \sin 6x + \cos 4x = -2 \), we start by analyzing the left-hand side of the equation.

### Step 1: Analyze the Range of the Functions
The sine and cosine functions have specific ranges:
- The range of \( \sin 6x \) is \([-1, 1]\).
- The range of \( \cos 4x \) is also \([-1, 1]\).

Thus, the sum \( \sin 6x + \cos 4x \) can take values from \(-2\) (when both functions are at their minimum) to \(2\) (when both functions are at their maximum).
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