`(sin theta" cosec "theta tan theta cot theta)/(sin^(2)theta+cos^(2)theta)`

Options

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

Answer

To solve the expression \((\sin \theta \cdot \csc \theta \cdot \tan \theta \cdot \cot \theta) / (\sin^2 \theta + \cos^2 \theta)\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Denominator**:
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Therefore, we can simplify the denominator:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]

2. **Rewrite the Expression**:
Substitute the denominator in the original expression:
\[ \frac{\sin \theta \cdot \csc \theta \cdot \tan \theta \cdot \cot \theta}{1} \]
This simplifies to:
\[ \sin \theta \cdot \csc \theta \cdot \tan \theta \cdot \cot \theta \]

3. **Use Trigonometric Identities**:
Recall the definitions of the trigonometric functions:
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

Substitute these identities into the expression:
\[ \sin \theta \cdot \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \]

4. **Simplify the Expression**:
Now, simplify the expression step by step:
- The \(\sin \theta\) in the numerator and denominator cancels out:
\[ 1 \cdot \frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \]
- The \(\sin \theta\) and \(\cos \theta\) also cancel out:
\[ 1 \cdot 1 = 1 \]

5. **Final Result**:
Therefore, the value of the original expression is:
\[ 1 \]

### Conclusion:
The final answer is \(1\).