What is the value of `(sin theta)/(1-cot theta) - (cos theta)/(1- tan theta)`

Options

A. `sin theta + cos theta `
B. `sin theta - cos theta`
C. `1//(sin theta + cos theta)`
D. `1//(sin theta - cos theta)`

Answer

To solve the expression \(\frac{\sin \theta}{1 - \cot \theta} - \frac{\cos \theta}{1 - \tan \theta}\), we will follow these steps:

### Step 1: Rewrite cotangent and tangent in terms of sine and cosine
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Using these definitions, we can rewrite the expression:
\[ \frac{\sin \theta}{1 - \cot \theta} = \frac{\sin \theta}{1 - \frac{\cos \theta}{\sin \theta}} = \frac{\sin \theta}{\frac{\sin \theta - \cos \theta}{\sin \theta}} = \frac{\sin^2 \theta}{\sin \theta - \cos \theta} \]
Similarly,
\[ \frac{\cos \theta}{1 - \tan \theta} = \frac{\cos \theta}{1 - \frac{\sin \theta}{\cos \theta}} = \frac{\cos \theta}{\frac{\cos \theta - \sin \theta}{\cos \theta}} = \frac{\cos^2 \theta}{\cos \theta - \sin \theta} \]

### Step 2: Substitute back into the expression
Now substituting these back into the original expression:
\[ \frac{\sin^2 \theta}{\sin \theta - \cos \theta} - \frac{\cos^2 \theta}{\cos \theta - \sin \theta} \]
Notice that \(\cos \theta - \sin \theta = -(\sin \theta - \cos \theta)\). Thus, we can rewrite the second term:
\[ \frac{\sin^2 \theta}{\sin \theta - \cos \theta} + \frac{\cos^2 \theta}{\sin \theta - \cos \theta} \]

### Step 3: Combine the fractions
Now we can combine the two fractions:
\[ \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta - \cos \theta} \]
Using the Pythagorean identity, we know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, the expression simplifies to:
\[ \frac{1}{\sin \theta - \cos \theta} \]

### Final Result
Therefore, the value of the expression is:
\[ \frac{1}{\sin \theta - \cos \theta} \]