What is `(costheta)/(1-tantheta)+(sin theta)/(1-cottheta)` equal to ?

Options

A. `sintheta-costheta`
B. `sintheta+costheta`
C. `2sintheta`
D. `2costheta`

Explanation

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

### Step 1: Rewrite \(\tan \theta\) and \(\cot \theta\)
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Substituting these into the expression gives:
\[ \frac{\cos \theta}{1 - \frac{\sin \theta}{\cos \theta}} + \frac{\sin \theta}{1 - \frac{\cos \theta}{\sin \theta}} \]

### Step 2: Simplify the denominators
The denominators can be simplified:
\[ 1 - \frac{\sin \theta}{\cos \theta} = \frac{\cos \theta - \sin \theta}{\cos \theta} \]
\[ 1 - \frac{\cos \theta}{\sin \theta} = \frac{\sin \theta - \cos \theta}{\sin \theta} \]
Thus, the expression becomes:
\[ \frac{\cos \theta}{\frac{\cos \theta - \sin \theta}{\cos \theta}} + \frac{\sin \theta}{\frac{\sin \theta - \cos \theta}{\sin \theta}} \]
This simplifies to:
\[ \frac{\cos^2 \theta}{\cos \theta - \sin \theta} + \frac{\sin^2 \theta}{\sin \theta - \cos \theta} \]

### Step 3: Combine the fractions
The second term can be rewritten by factoring out a negative sign:
\[ \frac{\sin^2 \theta}{\sin \theta - \cos \theta} = -\frac{\sin^2 \theta}{\cos \theta - \sin \theta} \]
Now we can combine the two fractions:
\[ \frac{\cos^2 \theta - \sin^2 \theta}{\cos \theta - \sin \theta} \]

### Step 4: Use the identity \(a^2 - b^2\)
Using the identity \(a^2 - b^2 = (a + b)(a - b)\):
\[ \cos^2 \theta - \sin^2 \theta = (\cos \theta + \sin \theta)(\cos \theta - \sin \theta) \]
Thus, we have:
\[ \frac{(\cos \theta + \sin \theta)(\cos \theta - \sin \theta)}{\cos \theta - \sin \theta} \]

### Step 5: Cancel out the common terms
Assuming \(\cos \theta \neq \sin \theta\), we can cancel \((\cos \theta - \sin \theta)\):
\[ \cos \theta + \sin \theta \]

### Final Answer
Thus, the expression \(\frac{\cos \theta}{1 - \tan \theta} + \frac{\sin \theta}{1 - \cot \theta}\) simplifies to:
\[ \cos \theta + \sin \theta \]
This matches option B: \(\sin \theta + \cos \theta\).