Find the general values of `theta ` which satisfies the equation ` tan theta =-1 and cos theta =(1)/(sqrt(2))`

Answer

To solve the equations \( \tan \theta = -1 \) and \( \cos \theta = \frac{1}{\sqrt{2}} \), we will find the general values of \( \theta \) that satisfy both conditions.

### Step 1: Solve \( \tan \theta = -1 \)

The tangent function is negative in the second and fourth quadrants. The principal value where \( \tan \theta = -1 \) is at \( \theta = -\frac{\pi}{4} \).

Using the general solution for the tangent function, we have:
\[ \theta = n\pi + \alpha \]
where \( \alpha = -\frac{\pi}{4} \) and \( n \) is any integer.

Thus, we can write:
\[ \theta = n\pi - \frac{\pi}{4} \]

### Step 2: Solve \( \cos \theta = \frac{1}{\sqrt{2}} \)

The cosine function is positive in the first and fourth quadrants. The principal value where \( \cos \theta = \frac{1}{\sqrt{2}} \) is at \( \theta = \frac{\pi}{4} \).

Using the general solution for the cosine function, we have:
\[ \theta = 2n\pi \pm \alpha \]
where \( \alpha = \frac{\pi}{4} \) and \( n \) is any integer.

Thus, we can write:
\[ \theta = 2n\pi + \frac{\pi}{4} \quad \text{or} \quad \theta = 2n\pi - \frac{\pi}{4} \]

### Step 3: Combine the solutions

Now we have two sets of solutions:
1. From \( \tan \theta = -1 \):
\[ \theta = n\pi - \frac{\pi}{4} \]
2. From \( \cos \theta = \frac{1}{\sqrt{2}} \):
\[ \theta = 2n\pi + \frac{\pi}{4} \quad \text{or} \quad \theta = 2n\pi - \frac{\pi}{4} \]

To find common values, we can set \( n\pi - \frac{\pi}{4} \) equal to either of the cosine solutions.

#### Case 1: Set \( n\pi - \frac{\pi}{4} = 2m\pi + \frac{\pi}{4} \)

Rearranging gives:
\[ n\pi - 2m\pi = \frac{\pi}{2} \]
\[ (n - 2m)\pi = \frac{\pi}{2} \]
\[ n - 2m = \frac{1}{2} \]
This does not yield integer solutions.

#### Case 2: Set \( n\pi - \frac{\pi}{4} = 2m\pi - \frac{\pi}{4} \)

Rearranging gives:
\[ n\pi = 2m\pi \]
\[ n = 2m \]
This yields integer solutions for \( n \) and \( m \).

### Final General Solutions

Thus, the general solutions for \( \theta \) that satisfy both equations are:
1. \( \theta = n\pi - \frac{\pi}{4} \) for \( n \) being any integer.
2. \( \theta = 2m\pi - \frac{\pi}{4} \) for \( m \) being any integer.