Find `("cosec" theta-sin theta)(sec theta-cos theta)(tan theta+cot theta)` :?

Options

A. `(3)/(4)`
B. `(2)/(3)`
C. `(2)/(5)`
D. `1`

Answer

To solve the expression \((\csc \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)\), we will break it down step by step.

### Step 1: Rewrite the trigonometric functions
We start by rewriting the trigonometric functions in terms of sine and cosine:
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

### Step 2: Substitute the values
Now we substitute these values into the expression:
\[ \left(\frac{1}{\sin \theta} - \sin \theta\right)\left(\frac{1}{\cos \theta} - \cos \theta\right)\left(\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}\right) \]

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

### Step 4: Combine the simplified terms
Now we combine all the simplified terms:
\[ \left(\frac{\cos^2 \theta}{\sin \theta}\right)\left(\frac{\sin^2 \theta}{\cos \theta}\right)\left(\frac{1}{\sin \theta \cos \theta}\right) \]

### Step 5: Multiply the fractions
Multiplying these fractions together:
\[ \frac{\cos^2 \theta \cdot \sin^2 \theta \cdot 1}{\sin \theta \cdot \cos \theta \cdot \sin \theta \cdot \cos \theta} = \frac{\cos^2 \theta \cdot \sin^2 \theta}{\sin^2 \theta \cdot \cos^2 \theta} = 1 \]

### Final Answer
Thus, the value of the expression \((\csc \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)\) is:
\[ \boxed{1} \]