The value of `("cosec" theta -cot theta)^(2)` is

Options

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

Answer

To find the value of \((\csc \theta - \cot \theta)^2\), we can follow these steps:

### Step 1: Rewrite \(\csc \theta\) and \(\cot \theta\)
We know that:
\[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
So, 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 over a common denominator:
\[ \csc \theta - \cot \theta = \frac{1 - \cos \theta}{\sin \theta} \]

### Step 3: Square the expression
Next, we need to square the entire expression:
\[ (\csc \theta - \cot \theta)^2 = \left(\frac{1 - \cos \theta}{\sin \theta}\right)^2 = \frac{(1 - \cos \theta)^2}{\sin^2 \theta} \]

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Thus, we can rewrite \(\sin^2 \theta\) in the denominator:
\[ (\csc \theta - \cot \theta)^2 = \frac{(1 - \cos \theta)^2}{1 - \cos^2 \theta} \]

### Step 5: Factor the denominator
The denominator can be factored as:
\[ 1 - \cos^2 \theta = (1 - \cos \theta)(1 + \cos \theta) \]
So, we have:
\[ (\csc \theta - \cot \theta)^2 = \frac{(1 - \cos \theta)^2}{(1 - \cos \theta)(1 + \cos \theta)} \]

### Step 6: Simplify the expression
Now, we can cancel one \((1 - \cos \theta)\) from the numerator and the denominator:
\[ (\csc \theta - \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta} \]

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

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