Choose the correct answer in each of the following questions:
`("cosec"theta-cot theta)^(2)=?`

Options

A. `(1+cos theta)/(1-cos theta)`
B. `(1-cos theta)/(1+cos theta)`
C. `(1+sin theta)/(1-sin theta)`
D. none of these

Explanation

To solve the expression \((\csc \theta - \cot \theta)^2\), we will follow these steps:

### Step 1: Rewrite the terms using sine and cosine
We know that:
\[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Thus, we can rewrite the expression:
\[ \csc \theta - \cot \theta = \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta} \]

### Step 2: Combine the fractions
Now, we can combine the fractions:
\[ \csc \theta - \cot \theta = \frac{1 - \cos \theta}{\sin \theta} \]

### Step 3: Square the expression
Now we will square the entire expression:
\[ (\csc \theta - \cot \theta)^2 = \left(\frac{1 - \cos \theta}{\sin \theta}\right)^2 \]
This gives us:
\[ = \frac{(1 - \cos \theta)^2}{\sin^2 \theta} \]

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \implies \sin^2 \theta = 1 - \cos^2 \theta \]
Thus, we can express \(\sin^2 \theta\) in terms of \(\cos \theta\):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]

### Step 5: Substitute and simplify
Now, substituting \(\sin^2 \theta\) into our expression:
\[ (\csc \theta - \cot \theta)^2 = \frac{(1 - \cos \theta)^2}{1 - \cos^2 \theta} \]
Notice that \(1 - \cos^2 \theta\) can be factored as \((1 - \cos \theta)(1 + \cos \theta)\):
\[ = \frac{(1 - \cos \theta)^2}{(1 - \cos \theta)(1 + \cos \theta)} \]

### Step 6: Cancel the common terms
We can cancel one \((1 - \cos \theta)\) from the numerator and the denominator:
\[ = \frac{1 - \cos \theta}{1 + \cos \theta} \]

### Final Result
Thus, the final answer is:
\[ (\csc \theta - \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta} \]