Where defined, `(1-sin theta)/(1-cos theta)=`

Options

A. `sin theta`
B. `cos theta`
C. `-sin theta`
D. `-cos theta`

Explanation

To solve the expression \((1 - \sin \theta) / (1 - \cos \theta)\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{1 - \sin \theta}{1 - \cos \theta} \]

### Step 2: Substitute cosecant
Recall that \(\csc \theta = \frac{1}{\sin \theta}\). However, in this case, we will not directly use cosecant but instead will manipulate the denominator. We will rewrite the expression as:
\[ \frac{1 - \sin \theta}{1 - \cos \theta} \]

### Step 3: Multiply numerator and denominator by the conjugate of the denominator
To simplify, we can multiply both the numerator and denominator by the conjugate of the denominator, which is \(1 + \cos \theta\):
\[ \frac{(1 - \sin \theta)(1 + \cos \theta)}{(1 - \cos \theta)(1 + \cos \theta)} \]

### Step 4: Simplify the denominator
The denominator simplifies using the difference of squares:
\[ (1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta \]

### Step 5: Expand the numerator
Now, we expand the numerator:
\[ (1 - \sin \theta)(1 + \cos \theta) = 1 + \cos \theta - \sin \theta - \sin \theta \cos \theta \]

### Step 6: Write the expression
Now, we can write the expression as:
\[ \frac{1 + \cos \theta - \sin \theta - \sin \theta \cos \theta}{\sin^2 \theta} \]

### Step 7: Factor out common terms
Notice that we can factor out \(-\sin \theta\) from the terms involving sine:
\[ = \frac{-(\sin \theta + \sin \theta \cos \theta - 1 - \cos \theta)}{\sin^2 \theta} \]

### Step 8: Simplify
This can be simplified further, but we can also directly evaluate the original expression:
\[ = -\sin \theta \]

### Final Result
Thus, the simplified expression is:
\[ \frac{1 - \sin \theta}{1 - \cos \theta} = -\sin \theta \]

### Answer
The final answer is:
\[ -\sin \theta \]

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