`(cot theta)/((1-sin theta)(sec theta + tan theta))` is equal to :

Options

A. cosec `theta`
B. sin `theta`
C. sec `theta`
D. 1

Answer

To solve the expression \(\frac{\cot \theta}{(1 - \sin \theta)(\sec \theta + \tan \theta)}\), we will simplify it step by step.

### Step 1: Rewrite cotangent, secant, and tangent
We know that:
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Substituting these into the expression gives us:
\[ \frac{\cot \theta}{(1 - \sin \theta)(\sec \theta + \tan \theta)} = \frac{\frac{\cos \theta}{\sin \theta}}{(1 - \sin \theta)\left(\frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta}\right)} \]

### Step 2: Simplify the denominator
The denominator can be simplified:
\[ \sec \theta + \tan \theta = \frac{1 + \sin \theta}{\cos \theta} \]
Thus, we have:
\[ (1 - \sin \theta)(\sec \theta + \tan \theta) = (1 - \sin \theta)\left(\frac{1 + \sin \theta}{\cos \theta}\right) \]

### Step 3: Combine the expression
Now, substituting this back into our expression gives:
\[ \frac{\frac{\cos \theta}{\sin \theta}}{(1 - \sin \theta)\left(\frac{1 + \sin \theta}{\cos \theta}\right)} = \frac{\cos \theta}{\sin \theta} \cdot \frac{\cos \theta}{(1 - \sin \theta)(1 + \sin \theta)} \]

### Step 4: Simplify further
The expression simplifies to:
\[ \frac{\cos^2 \theta}{\sin \theta (1 - \sin^2 \theta)} \]
Using the identity \(1 - \sin^2 \theta = \cos^2 \theta\), we can rewrite this as:
\[ \frac{\cos^2 \theta}{\sin \theta \cos^2 \theta} \]

### Step 5: Final simplification
This simplifies to:
\[ \frac{1}{\sin \theta} \]
Which is equal to \(\csc \theta\).

### Conclusion
Thus, the final answer is:
\[ \csc \theta \]