Find the most general value of ` theta` satisfyingn the equation `tantheta=-1 and costheta=(1 )/(sqrt(2))`.

Options

A. `theta= 2npi+(7pi)/(4)`.
B. `theta= npi+(7pi)/(4)`.
C. `theta= 2npi+(pi)/(4)`.
D. none of these

Explanation

To solve the problem of finding the most general value of \( \theta \) satisfying the equations \( \tan \theta = -1 \) and \( \cos \theta = \frac{1}{\sqrt{2}} \), we can follow these steps:

### Step 1: Analyze the equation \( \tan \theta = -1 \)
The tangent function is negative in the second and fourth quadrants. The specific angles where \( \tan \theta = -1 \) are:
- \( \theta = \frac{3\pi}{4} \) (in the second quadrant)
- \( \theta = \frac{7\pi}{4} \) (in the fourth quadrant)

### Step 2: Write the general solutions for \( \tan \theta = -1 \)
The general solutions for \( \tan \theta = -1 \) can be expressed as:
\[ \theta = n\pi - \frac{\pi}{4} \]
where \( n \) is any integer.

### Step 3: Analyze the equation \( \cos \theta = \frac{1}{\sqrt{2}} \)
The cosine function is positive in the first and fourth quadrants. The specific angles where \( \cos \theta = \frac{1}{\sqrt{2}} \) are:
- \( \theta = \frac{\pi}{4} \) (in the first quadrant)
- \( \theta = \frac{7\pi}{4} \) (in the fourth quadrant)

### Step 4: Write the general solutions for \( \cos \theta = \frac{1}{\sqrt{2}} \)
The general solutions for \( \cos \theta = \frac{1}{\sqrt{2}} \) can be expressed as:
\[ \theta = 2n\pi \pm \frac{\pi}{4} \]
where \( n \) is any integer.

### Step 5: Find the common solutions
Now, we need to find the common solutions from both equations:
1. From \( \tan \theta = -1 \):
\[ \theta = n\pi - \frac{\pi}{4} \]
2. From \( \cos \theta = \frac{1}{\sqrt{2}} \):
\[ \theta = 2n\pi + \frac{\pi}{4} \quad \text{and} \quad \theta = 2n\pi - \frac{\pi}{4} \]

The common solution from both sets is:
- For \( \theta = 2n\pi - \frac{\pi}{4} \) (from cosine) and \( \theta = n\pi - \frac{\pi}{4} \) (from tangent), we can see that both share the term \( -\frac{\pi}{4} \).

### Step 6: Write the final general solution
Thus, the most general values of \( \theta \) satisfying both equations can be expressed as:
\[ \theta = 2n\pi - \frac{\pi}{4} \quad \text{and} \quad \theta = 2n\pi + \frac{7\pi}{4} \]
where \( n \) is any integer.

### Final Answer:
The most general values of \( \theta \) are:
\[ \theta = 2n\pi - \frac{\pi}{4} \quad \text{and} \quad \theta = 2n\pi + \frac{7\pi}{4} \]