Find the value of
` (cos theta)/((1-tan theta))+(sin theta)/((1-cot theta))`

Options

A. `cos theta + sin theta`
B. `cos theta - sin theta`
C. `1/(cos theta + sin theta)`
D. `(cos theta + sin theta)/2`

Explanation

To solve the expression

\[ \frac{\cos \theta}{1 - \tan \theta} + \frac{\sin \theta}{1 - \cot \theta} \]

we will simplify each term step by step.

### Step 1: Rewrite \(\tan \theta\) and \(\cot \theta\)

We know that:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]

### Step 2: Substitute \(\tan \theta\) and \(\cot \theta\)

Substituting these definitions into the expression gives:

\[ \frac{\cos \theta}{1 - \frac{\sin \theta}{\cos \theta}} + \frac{\sin \theta}{1 - \frac{\cos \theta}{\sin \theta}} \]

### Step 3: Simplify the denominators

For the first term:

\[ 1 - \frac{\sin \theta}{\cos \theta} = \frac{\cos \theta - \sin \theta}{\cos \theta} \]

Thus, the first term becomes:

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

For the second term:

\[ 1 - \frac{\cos \theta}{\sin \theta} = \frac{\sin \theta - \cos \theta}{\sin \theta} \]

Thus, the second term becomes:

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

### Step 4: Combine the two terms

Now we have:

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

Notice that \(\sin \theta - \cos \theta = -(\cos \theta - \sin \theta)\). Thus, we can rewrite the second term:

\[ \frac{\sin^2 \theta}{-(\cos \theta - \sin \theta)} = -\frac{\sin^2 \theta}{\cos \theta - \sin \theta} \]

Combining both terms gives:

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

### Step 5: Factor the numerator

The numerator can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\):

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

### Step 6: Cancel common factors

Now substituting back, we have:

\[ \frac{(\cos \theta - \sin \theta)(\cos \theta + \sin \theta)}{\cos \theta - \sin \theta} \]

Assuming \(\cos \theta \neq \sin \theta\), we can cancel \((\cos \theta - \sin \theta)\):

\[ \cos \theta + \sin \theta \]

### Conclusion

Thus, we have shown that:

\[ \frac{\cos \theta}{1 - \tan \theta} + \frac{\sin \theta}{1 - \cot \theta} = \cos \theta + \sin \theta \]