Find all the number 'a' for which any root of the equation ` sin 3x =a sin x +(4-2|a|)sin^(2) x ` is a root of the equation ` sin 3x + cos 2 x =1+2 sinx cos 2x ` and any root of the latter equation is a root of the former .

Answer

To solve the problem, we need to find all values of \( a \) such that any root of the equation

\[ \sin 3x = a \sin x + (4 - 2|a|) \sin^2 x \]

is also a root of the equation

\[ \sin 3x + \cos 2x = 1 + 2 \sin x \cos 2x \]

and vice versa.

### Step-by-step Solution:

1. **Analyze the second equation:**
We start with the second equation:

\[ \sin 3x + \cos 2x = 1 + 2 \sin x \cos 2x \]

We can rewrite \( 2 \sin x \cos 2x \) using the sine addition formula:

\[ 2 \sin x \cos 2x = \sin(x + 2x) + \sin(x - 2x) = \sin(3x) + \sin(-x) \]

Therefore, the equation simplifies to:

\[ \sin 3x + \cos 2x = 1 + \sin 3x - \sin x \]

Cancelling \( \sin 3x \) from both sides gives:

\[ \cos 2x = 1 - \sin x \]

Using the identity \( \cos 2x = 1 - 2 \sin^2 x \), we have:

\[ 1 - 2 \sin^2 x = 1 - \sin x \]

Simplifying leads to:

\[ 2 \sin^2 x - \sin x = 0 \]

Factoring out \( \sin x \):

\[ \sin x (2 \sin x - 1) = 0 \]

This gives us the roots:

\[ \sin x = 0 \quad \text{or} \quad \sin x = \frac{1}{2} \]

2. **Analyze the first equation:**
Now, we consider the first equation:

\[ \sin 3x = a \sin x + (4 - 2|a|) \sin^2 x \]

Rewriting \( \sin 3x \) using the triple angle formula gives:

\[ 3 \sin x - 4 \sin^3 x = a \sin x + (4 - 2|a|) \sin^2 x \]

Rearranging leads to:

\[ 3 \sin x - a \sin x - (4 - 2|a|) \sin^2 x - 4 \sin^3 x = 0 \]

Factoring out \( \sin x \):

\[ \sin x (3 - a - (4 - 2|a|) \sin x - 4 \sin^2 x) = 0 \]

This gives us one root \( \sin x = 0 \). The other roots come from solving:

\[ 4 \sin^3 x + (4 - 2|a|) \sin^2 x + (a - 3) = 0 \]

3. **Matching roots:**
We need the second equation's roots \( \sin x = 0 \) and \( \sin x = \frac{1}{2} \) to satisfy the cubic equation.

For \( \sin x = \frac{1}{2} \):

\[ 4 \left(\frac{1}{2}\right)^3 + (4 - 2|a|) \left(\frac{1}{2}\right)^2 + (a - 3) = 0 \]

Simplifying gives:

\[ 4 \cdot \frac{1}{8} + (4 - 2|a|) \cdot \frac{1}{4} + a - 3 = 0 \]

This leads to:

\[ \frac{1}{2} + (1 - \frac{|a|}{2}) + a - 3 = 0 \]

Rearranging gives:

\[ a - \frac{|a|}{2} - \frac{5}{2} = 0 \]

This can be solved for \( a \).

4. **Finding conditions on \( a \):**
From the equation \( a - \frac{|a|}{2} = \frac{5}{2} \), we analyze two cases based on the sign of \( a \):

- If \( a \geq 0 \):
\[ a - \frac{a}{2} = \frac{5}{2} \implies \frac{a}{2} = \frac{5}{2} \implies a = 5 \]

- If \( a < 0 \):
\[ a + \frac{a}{2} = \frac{5}{2} \implies \frac{3a}{2} = \frac{5}{2} \implies a = \frac{5}{3} \]

5. **Final conditions:**
We also require \( a \) to satisfy \( |a - 3| \leq 2 \):

\[ 1 \leq a \leq 5 \]

Thus, the values of \( a \) that satisfy both conditions are:

\[ a = 3, \quad a = 4, \quad \text{and} \quad 1 \leq a \leq 5 \]

### Final Answer:
The values of \( a \) are \( a = 3, a = 4 \), and \( a \in [1, 5] \).