What is the value of
`("cosec "(pi+theta)cot{(9pi//2-theta)}"cosec"^(2)(2pi-theta))/(cot(2pi-theta)sec^(2)(pi-theta)sec{(3pi//2)+theta})`

Options

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

Explanation

To solve the expression

\[ \frac{\text{cosec}(\pi + \theta) \cdot \cot\left(9\frac{\pi}{2} - \theta\right) \cdot \text{cosec}^2(2\pi - \theta)}{\cot(2\pi - \theta) \cdot \sec^2(\pi - \theta) \cdot \sec\left(3\frac{\pi}{2} + \theta\right)} \]

we will evaluate each term step by step.

### Step 1: Evaluate \(\text{cosec}(\pi + \theta)\)

Using the identity \(\text{cosec}(\pi + \theta) = -\text{cosec}(\theta)\):

\[ \text{cosec}(\pi + \theta) = -\text{cosec}(\theta) \]

### Step 2: Evaluate \(\cot\left(9\frac{\pi}{2} - \theta\right)\)

We can rewrite \(9\frac{\pi}{2}\) as \(4\pi + \frac{\pi}{2}\). Since \(\cot\) has a period of \(\pi\):

\[ \cot\left(9\frac{\pi}{2} - \theta\right) = \cot\left(\frac{\pi}{2} - \theta\right) = \tan(\theta) \]

### Step 3: Evaluate \(\text{cosec}^2(2\pi - \theta)\)

Using the identity \(\text{cosec}(2\pi - \theta) = \text{cosec}(\theta)\):

\[ \text{cosec}^2(2\pi - \theta) = \text{cosec}^2(\theta) \]

### Step 4: Evaluate \(\cot(2\pi - \theta)\)

Using the identity \(\cot(2\pi - \theta) = -\cot(\theta)\):

\[ \cot(2\pi - \theta) = -\cot(\theta) \]

### Step 5: Evaluate \(\sec^2(\pi - \theta)\)

Using the identity \(\sec(\pi - \theta) = -\sec(\theta)\):

\[ \sec^2(\pi - \theta) = \sec^2(\theta) \]

### Step 6: Evaluate \(\sec\left(3\frac{\pi}{2} + \theta\right)\)

Using the identity \(\sec\left(3\frac{\pi}{2} + \theta\right) = -\csc(\theta)\):

\[ \sec\left(3\frac{\pi}{2} + \theta\right) = -\csc(\theta) \]

### Step 7: Substitute all evaluated terms into the expression

Now substituting all the evaluated terms into the original expression:

\[ \frac{-\text{cosec}(\theta) \cdot \tan(\theta) \cdot \text{cosec}^2(\theta)}{-\cot(\theta) \cdot \sec^2(\theta) \cdot (-\csc(\theta))} \]

### Step 8: Simplify the expression

The expression simplifies to:

\[ \frac{\text{cosec}(\theta) \cdot \tan(\theta) \cdot \text{cosec}^2(\theta)}{\cot(\theta) \cdot \sec^2(\theta) \cdot \csc(\theta)} \]

### Step 9: Rewrite \(\tan\) and \(\cot\)

Recall that:

\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \quad \text{and} \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]

Substituting these into the expression gives:

\[ \frac{\text{cosec}^3(\theta) \cdot \frac{\sin(\theta)}{\cos(\theta)}}{\frac{\cos(\theta)}{\sin(\theta)} \cdot \sec^2(\theta) \cdot \csc(\theta)} \]

### Step 10: Further simplification

This simplifies to:

\[ \frac{\text{cosec}^3(\theta) \cdot \sin(\theta)}{\frac{\cos^2(\theta)}{\sin(\theta)} \cdot \sec^2(\theta)} \]

### Step 11: Final simplification

This results in:

\[ \frac{\text{cosec}^3(\theta) \cdot \sin^2(\theta)}{\cos^2(\theta)} = \tan^2(\theta) \cdot \text{cosec}^2(\theta) \]

### Step 12: Conclusion

Since \(\tan^2(\theta) \cdot \text{cosec}^2(\theta) = 1\), the final value of the expression is:

\[ \boxed{1} \]