Compute
`cosec (13pi//12)`

Answer

To compute \( \csc\left(\frac{13\pi}{12}\right) \), we will follow these steps:

### Step 1: Simplify the angle
We can express \( \frac{13\pi}{12} \) in a more manageable form:
\[ \frac{13\pi}{12} = \pi + \frac{\pi}{12} \]
This means we can rewrite \( \csc\left(\frac{13\pi}{12}\right) \) as:
\[ \csc\left(\frac{13\pi}{12}\right) = \csc\left(\pi + \frac{\pi}{12}\right) \]

### Step 2: Use the cosecant identity
Using the identity for cosecant, we know:
\[ \csc(\pi + \theta) = -\csc(\theta) \]
Thus, we have:
\[ \csc\left(\frac{13\pi}{12}\right) = -\csc\left(\frac{\pi}{12}\right) \]

### Step 3: Find \( \csc\left(\frac{\pi}{12}\right) \)
Now we need to find \( \csc\left(\frac{\pi}{12}\right) \), which is the reciprocal of \( \sin\left(\frac{\pi}{12}\right) \):
\[ \csc\left(\frac{\pi}{12}\right) = \frac{1}{\sin\left(\frac{\pi}{12}\right)} \]

### Step 4: Calculate \( \sin\left(\frac{\pi}{12}\right) \)
We can find \( \sin\left(\frac{\pi}{12}\right) \) using the sine subtraction formula:
\[ \sin\left(\frac{\pi}{12}\right) = \sin\left(15^\circ\right) = \sin(45^\circ - 30^\circ) \]
Using the sine subtraction formula:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
Let \( a = 45^\circ \) and \( b = 30^\circ \):
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \cos(45^\circ) = \frac{1}{\sqrt{2}}, \quad \sin(30^\circ) = \frac{1}{2} \]
Substituting these values:
\[ \sin\left(15^\circ\right) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{1}{2} \]
\[ = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Step 5: Substitute back to find \( \csc\left(\frac{\pi}{12}\right) \)
Now substituting back:
\[ \csc\left(\frac{\pi}{12}\right) = \frac{1}{\sin\left(\frac{\pi}{12}\right)} = \frac{2\sqrt{2}}{\sqrt{3} - 1} \]

### Step 6: Rationalize the denominator
To rationalize the denominator:
\[ \csc\left(\frac{\pi}{12}\right) = \frac{2\sqrt{2}(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{2\sqrt{2}(\sqrt{3} + 1)}{3 - 1} = \frac{2\sqrt{2}(\sqrt{3} + 1)}{2} = \sqrt{2}(\sqrt{3} + 1) \]

### Step 7: Final substitution
Now substituting back to find \( \csc\left(\frac{13\pi}{12}\right) \):
\[ \csc\left(\frac{13\pi}{12}\right) = -\csc\left(\frac{\pi}{12}\right) = -\sqrt{2}(\sqrt{3} + 1) \]

### Final Answer
Thus, the final answer is:
\[ \csc\left(\frac{13\pi}{12}\right) = -\sqrt{2}(\sqrt{3} + 1) \]

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