The number of distinct real roots of the equation `sin^(3)x +sin^(2)x + sin x-2cosxsin^2 x- sin 2x-2cos x=0` belonging to the interval `(-(pi)/(2),(pi)/(2))` is

Options

A. (a) 0
B. (b) 1
C. (c) 2
D. (d) 3

Explanation

To find the number of distinct real roots of the equation

\[ \sin^3 x + \sin^2 x + \sin x - 2 \cos x \sin^2 x - \sin 2x - 2 \cos x = 0 \]

in the interval \((- \frac{\pi}{2}, \frac{\pi}{2})\), we will simplify the equation step by step.

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

\[ \sin^3 x + \sin^2 x + \sin x - 2 \cos x \sin^2 x - \sin 2x - 2 \cos x = 0 \]

### Step 2: Substitute \(\sin 2x\)
Recall that \(\sin 2x = 2 \sin x \cos x\). We can substitute this into the equation:

\[ \sin^3 x + \sin^2 x + \sin x - 2 \cos x \sin^2 x - 2 \sin x \cos x - 2 \cos x = 0 \]

### Step 3: Combine like terms
Now, we can group the terms:

\[ \sin^3 x + \sin^2 x + \sin x - 2 \cos x (\sin^2 x + \sin x + 1) = 0 \]

### Step 4: Factor out common terms
We can factor out \((\sin x - 2 \cos x)\):

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

### Step 5: Analyze the factors
This gives us two equations to solve:

1. \(\sin x - 2 \cos x = 0\)
2. \(\sin^2 x + \sin x + 1 = 0\)

### Step 6: Solve the first equation
From \(\sin x - 2 \cos x = 0\), we can rearrange it to:

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

Dividing both sides by \(\cos x\) (valid since \(\cos x \neq 0\) in the interval \((- \frac{\pi}{2}, \frac{\pi}{2})\)) gives:

\[ \tan x = 2 \]

Thus,

\[ x = \tan^{-1}(2) \]

### Step 7: Solve the second equation
Now, we analyze the second equation:

\[ \sin^2 x + \sin x + 1 = 0 \]

The discriminant of this quadratic equation is:

\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]

Since the discriminant is negative, this equation has no real roots.

### Conclusion
Thus, the only real root in the interval \((- \frac{\pi}{2}, \frac{\pi}{2})\) comes from the first equation:

\[ x = \tan^{-1}(2) \]

Therefore, the number of distinct real roots of the given equation in the specified interval is:

\[ \boxed{1} \]