What is the value of `(sectheta-costheta)("cosec"theta-sintheta)(cottheta+tantheta)`?

Options

A. 1
B. 2
C. `1sintheta`
D. `costheta`

Explanation

To find the value of the expression \((\sec \theta - \cos \theta)(\csc \theta - \sin \theta)(\cot \theta + \tan \theta)\), we will simplify each part step by step.

### Step 1: Rewrite the trigonometric functions
We start by rewriting the trigonometric functions in terms of sine and cosine.

\[ \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]

### Step 2: Substitute the rewritten functions into the expression
Now we substitute these definitions into the expression:

\[ \left(\frac{1}{\cos \theta} - \cos \theta\right)\left(\frac{1}{\sin \theta} - \sin \theta\right)\left(\frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta}\right) \]

### Step 3: Simplify each term
1. **First term**:
\[ \frac{1}{\cos \theta} - \cos \theta = \frac{1 - \cos^2 \theta}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta} \]
(since \(1 - \cos^2 \theta = \sin^2 \theta\))

2. **Second term**:
\[ \frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta} \]
(since \(1 - \sin^2 \theta = \cos^2 \theta\))

3. **Third term**:
\[ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} = \cot \theta + \tan \theta = \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \]
(since \(\cos^2 \theta + \sin^2 \theta = 1\))

### Step 4: Combine the simplified terms
Now we substitute these simplified terms back into the expression:

\[ \left(\frac{\sin^2 \theta}{\cos \theta}\right)\left(\frac{\cos^2 \theta}{\sin \theta}\right)\left(\frac{1}{\sin \theta \cos \theta}\right) \]

### Step 5: Simplify the entire expression
Now we can multiply these fractions:

\[ \frac{\sin^2 \theta \cdot \cos^2 \theta \cdot 1}{\cos \theta \cdot \sin \theta \cdot \sin \theta \cdot \cos \theta} = \frac{\sin^2 \theta \cdot \cos^2 \theta}{\sin^2 \theta \cdot \cos^2 \theta} = 1 \]

### Final Result
Thus, the value of the expression \((\sec \theta - \cos \theta)(\csc \theta - \sin \theta)(\cot \theta + \tan \theta)\) is:

\[ \boxed{1} \]