The number of solutions of the equation
` (2 sin((sin x )/(2)))(cos((sinx)/(2)))(sin(2 "tan" (x)/(2) " cos"^(2) (x)/(2))-3) + 2 = 0 ` in ` [0, 2 pi ] ` is :

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the equation

\[ (2 \sin(\frac{\sin x}{2}))(\cos(\frac{\sin x}{2}))\left(\sin\left(2 \tan\left(\frac{x}{2}\right)\right) \cos^2\left(\frac{x}{2}\right) - 3\right) + 2 = 0 \]

in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation for clarity:

\[ 2 \sin\left(\frac{\sin x}{2}\right) \cos\left(\frac{\sin x}{2}\right) \left(\sin\left(2 \tan\left(\frac{x}{2}\right)\right) \cos^2\left(\frac{x}{2}\right) - 3\right) + 2 = 0 \]

### Step 2: Simplify using trigonometric identities
Using the identity \(2 \sin \theta \cos \theta = \sin(2\theta)\), we can simplify the first part:

\[ \sin\left(\sin x\right) \left(\sin\left(2 \tan\left(\frac{x}{2}\right)\right) \cos^2\left(\frac{x}{2}\right) - 3\right) + 2 = 0 \]

### Step 3: Set the equation to zero
We can set the equation to zero:

\[ \sin\left(\sin x\right) \left(\sin\left(2 \tan\left(\frac{x}{2}\right)\right) \cos^2\left(\frac{x}{2}\right) - 3\right) = -2 \]

### Step 4: Analyze the components
We analyze the equation:

1. The term \(\sin\left(\sin x\right)\) varies between \(-1\) and \(1\).
2. The term \(\sin\left(2 \tan\left(\frac{x}{2}\right)\right) \cos^2\left(\frac{x}{2}\right) - 3\) can be evaluated for different values of \(x\).

### Step 5: Find the range of the components
Since \(\sin\left(\sin x\right)\) can only take values from \(-1\) to \(1\), the left-hand side cannot equal \(-2\). Therefore, there are no values of \(x\) that satisfy the equation.

### Conclusion
The number of solutions of the equation in the interval \([0, 2\pi]\) is:

\[ \text{Number of solutions} = 0 \]