If `sin theta sec theta =-1` and `theta` lies in the second quadrant, find `sin theta` and `sec theta`.

Answer

To solve the problem where \( \sin \theta \sec \theta = -1 \) and \( \theta \) lies in the second quadrant, we can follow these steps:

### Step 1: Rewrite the equation
Given:
\[ \sin \theta \sec \theta = -1 \]
We know that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, we can rewrite the equation as:
\[ \sin \theta \cdot \frac{1}{\cos \theta} = -1 \]
This simplifies to:
\[ \frac{\sin \theta}{\cos \theta} = -1 \]

### Step 2: Recognize the tangent function
The equation \( \frac{\sin \theta}{\cos \theta} = -1 \) can be expressed as:
\[ \tan \theta = -1 \]

### Step 3: Determine the angle in the second quadrant
Since \( \theta \) lies in the second quadrant and \( \tan \theta = -1 \), we can find the angle:
\[ \tan \theta = \tan\left(\frac{3\pi}{4}\right) \]
Thus, we have:
\[ \theta = \frac{3\pi}{4} \]

### Step 4: Calculate \( \sin \theta \)
Now, we find \( \sin \theta \) at \( \theta = \frac{3\pi}{4} \):
\[ \sin\left(\frac{3\pi}{4}\right) = \sin\left(\pi - \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 5: Calculate \( \sec \theta \)
Next, we calculate \( \sec \theta \):
\[ \sec\left(\frac{3\pi}{4}\right) = \frac{1}{\cos\left(\frac{3\pi}{4}\right)} = \frac{1}{-\cos\left(\frac{\pi}{4}\right)} = -\frac{1}{\frac{1}{\sqrt{2}}} = -\sqrt{2} \]

### Conclusion
Thus, we find:
\[ \sin \theta = \frac{1}{\sqrt{2}}, \quad \sec \theta = -\sqrt{2} \]

### Summary of Results
- \( \sin \theta = \frac{1}{\sqrt{2}} \)
- \( \sec \theta = -\sqrt{2} \)