`(cosec theta + sin theta) (sec theta - cos theta) tan theta + cot theta)` is equal to

Options

A. 0
B. 1
C. `-1`
D. None of these

Explanation

To solve the expression \((\csc \theta + \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)\), we will convert all trigonometric functions into sine and cosine.

### Step 1: Rewrite the expression using sine and cosine
We start by rewriting each term:
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

Thus, the expression becomes:
\[ \left(\frac{1}{\sin \theta} + \sin \theta\right)\left(\frac{1}{\cos \theta} - \cos \theta\right)\left(\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}\right) \]

### Step 2: Simplify the first bracket
The first bracket simplifies as follows:
\[ \frac{1 + \sin^2 \theta}{\sin \theta} \]

### Step 3: Simplify the second bracket
The second bracket simplifies as follows:
\[ \frac{1 - \cos^2 \theta}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta} \]

### Step 4: Simplify the third bracket
The third bracket simplifies as follows:
\[ \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \]

### Step 5: Combine all parts
Now we combine all the simplified parts:
\[ \left(\frac{1 + \sin^2 \theta}{\sin \theta}\right) \left(\frac{\sin^2 \theta}{\cos \theta}\right) \left(\frac{1}{\sin \theta \cos \theta}\right) \]

### Step 6: Multiply the fractions
Now we multiply all these fractions together:
\[ \frac{(1 + \sin^2 \theta) \cdot \sin^2 \theta}{\sin^2 \theta \cdot \cos^2 \theta} \]

### Step 7: Cancel out terms
The \(\sin^2 \theta\) in the numerator and denominator cancels out:
\[ \frac{1 + \sin^2 \theta}{\cos^2 \theta} \]

### Step 8: Final simplification
This can be rewritten as:
\[ \sec^2 \theta + \tan^2 \theta \]

Using the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), we can conclude that:
\[ \sec^2 \theta + \tan^2 \theta = 2\tan^2 \theta + 1 \]

### Final Answer:
Thus, the expression simplifies to:
\[ 2 \]