If `(sin^(3)theta-cos^(3)theta)/(sin theta-cos theta)-(cos theta)/(sqrt(1+cot^(2)theta))-2 tan theta cot theta=-1 (AA theta in[0, 2pi],` then

Options

A. `theta in (0, (pi)/(2))-{(pi)/(4)}`
B. `theta in ((pi)/(2),pi)-{(3pi)/(4)}`
C. `theta in (pi, (3pi)/(2))-{(5pi)/(4)}`
D. `theta in (0, pi)-{(pi)/(4), (pi)/(2)}`

Answer

To solve the equation

\[ \frac{\sin^3 \theta - \cos^3 \theta}{\sin \theta - \cos \theta} - \frac{\cos \theta}{\sqrt{1 + \cot^2 \theta}} - 2 \tan \theta \cot \theta = -1 \]

we will follow these steps:

### Step 1: Simplify the first term

Using the identity for the difference of cubes, we can rewrite \(\sin^3 \theta - \cos^3 \theta\) as:

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

Thus, we can simplify the first term:

\[ \frac{\sin^3 \theta - \cos^3 \theta}{\sin \theta - \cos \theta} = \sin^2 \theta + \sin \theta \cos \theta + \cos^2 \theta \]

Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have:

\[ \sin^2 \theta + \sin \theta \cos \theta + \cos^2 \theta = 1 + \sin \theta \cos \theta \]

### Step 2: Simplify the second term

The term \(\sqrt{1 + \cot^2 \theta}\) can be simplified using the identity \(1 + \cot^2 \theta = \csc^2 \theta\):

\[ \sqrt{1 + \cot^2 \theta} = \csc \theta \]

Thus, the second term becomes:

\[ -\frac{\cos \theta}{\csc \theta} = -\cos \theta \cdot \sin \theta = -\sin \theta \cos \theta \]

### Step 3: Substitute back into the equation

Now substituting these simplifications back into the original equation gives us:

\[ 1 + \sin \theta \cos \theta - \sin \theta \cos \theta - 2 \tan \theta \cot \theta = -1 \]

Since \(\tan \theta \cot \theta = 1\):

\[ 1 - 2 = -1 \]

This simplifies to:

\[ -1 = -1 \]

### Step 4: Analyze the equation

The equation holds true for all values of \(\theta\) in the interval \([0, 2\pi]\). However, we need to check for any restrictions or specific values where the terms might be undefined.

### Step 5: Check for undefined points

The term \(\tan \theta\) and \(\cot \theta\) can be undefined at \(\theta = k\pi\) for integer \(k\). Specifically, we need to avoid:

- \(\theta = 0\)
- \(\theta = \pi\)
- \(\theta = 2\pi\)

### Conclusion

Thus, the values of \(\theta\) that satisfy the equation in the interval \([0, 2\pi]\) are:

\[ \theta \in (0, 2\pi) \setminus \{0, \pi, 2\pi\} \]