`(cosectheta-sintheta)(sectheta-costheta)(tantheta+cottheta)=?`

Options

A. `-1`
B. 1
C. 2
D. 0

Answer

To simplify the expression \((\csc \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)\), we will follow these steps:

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

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 each bracket
1. For \(\frac{1}{\sin \theta} - \sin \theta\):
\[ = \frac{1 - \sin^2 \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta} \]

2. For \(\frac{1}{\cos \theta} - \cos \theta\):
\[ = \frac{1 - \cos^2 \theta}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta} \]

3. For \(\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}\):
\[ = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \]

### Step 3: Combine the simplified parts
Now we substitute back into the expression:
\[ \left(\frac{\cos^2 \theta}{\sin \theta}\right)\left(\frac{\sin^2 \theta}{\cos \theta}\right)\left(\frac{1}{\sin \theta \cos \theta}\right) \]

### Step 4: Multiply the fractions
Combining these fractions:
\[ = \frac{\cos^2 \theta \cdot \sin^2 \theta \cdot 1}{\sin \theta \cdot \cos \theta \cdot \sin \theta \cdot \cos \theta} \]
\[ = \frac{\cos^2 \theta \sin^2 \theta}{\sin^2 \theta \cos^2 \theta} \]

### Step 5: Simplify the expression
The \(\sin^2 \theta\) and \(\cos^2 \theta\) in the numerator and denominator cancel out:
\[ = 1 \]

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