Number of solutions of the equation `cos 5x xx tan (6|x|)+sin 5x=0` lying in `[-2pi, pi)` is _____.

Explanation

To find the number of solutions of the equation \( \cos(5x) \tan(6|x|) + \sin(5x) = 0 \) in the interval \([-2\pi, \pi)\), we will consider two cases based on the absolute value of \(x\).

### Step 1: Case 1 - \(x \geq 0\)

In this case, \(|x| = x\). The equation becomes:
\[ \cos(5x) \tan(6x) + \sin(5x) = 0 \]
We can express \(\tan(6x)\) as \(\frac{\sin(6x)}{\cos(6x)}\):
\[ \cos(5x) \frac{\sin(6x)}{\cos(6x)} + \sin(5x) = 0 \]
Multiplying through by \(\cos(6x)\) (assuming \(\cos(6x) \neq 0\)):
\[ \cos(5x) \sin(6x) + \sin(5x) \cos(6x) = 0 \]
This can be rewritten using the sine addition formula:
\[ \sin(5x + 6x) = 0 \]
which simplifies to:
\[ \sin(11x) = 0 \]
The solutions to this equation are given by:
\[ 11x = n\pi \quad \Rightarrow \quad x = \frac{n\pi}{11} \]
Now, we need to find the values of \(n\) such that \(x\) lies in the interval \([-2\pi, \pi)\).

Calculating the limits for \(n\):
- For \(x = -2\pi\):
\[ -2\pi \leq \frac{n\pi}{11} \quad \Rightarrow \quad n \geq -22 \]
- For \(x = \pi\):
\[ \frac{n\pi}{11} < \pi \quad \Rightarrow \quad n < 11 \]

Thus, \(n\) can take values from \(-22\) to \(10\), which gives:
\[ n = -22, -21, \ldots, 10 \]
Counting these values:
\[ 10 - (-22) + 1 = 33 \text{ solutions} \]

### Step 2: Case 2 - \(x < 0\)

In this case, \(|x| = -x\). The equation becomes:
\[ \cos(5x) \tan(-6x) + \sin(5x) = 0 \]
Using the property \(\tan(-\theta) = -\tan(\theta)\):
\[ \cos(5x)(-\tan(6x)) + \sin(5x) = 0 \quad \Rightarrow \quad -\cos(5x) \tan(6x) + \sin(5x) = 0 \]
This can be rearranged to:
\[ \sin(5x) = \cos(5x) \tan(6x) \]
Following similar steps as before, we can express this as:
\[ \sin(5x - 6x) = 0 \quad \Rightarrow \quad \sin(-x) = 0 \]
This gives:
\[ -x = n\pi \quad \Rightarrow \quad x = -n\pi \]
We need to find values of \(n\) such that \(x\) lies in the interval \([-2\pi, 0)\):
- For \(x = -2\pi\):
\[ -2\pi < -n\pi \quad \Rightarrow \quad n < 2 \]
- For \(x = 0\):
\[ -n\pi < 0 \quad \Rightarrow \quad n > 0 \]

Thus, \(n\) can take values \(1\) (since \(n\) must be a positive integer). This gives us:
\[ n = 1 \]
So there is \(1\) solution from this case.

### Final Count of Solutions

Adding the solutions from both cases:
\[ 33 \text{ (from case 1)} + 1 \text{ (from case 2)} = 34 \]

Thus, the total number of solutions of the equation \( \cos(5x) \tan(6|x|) + \sin(5x) = 0 \) in the interval \([-2\pi, \pi)\) is:
\[ \boxed{34} \]