If tan ` theta = 1 and sin phi = (1)/( sqrt(2))` , then the value of ` cos (theta + phi)` is

Options

A. `-1`
B. 0
C. 1
D. `(sqrt(3))/(2)`

Answer

To find the value of \( \cos(\theta + \phi) \) given that \( \tan \theta = 1 \) and \( \sin \phi = \frac{1}{\sqrt{2}} \), we can follow these steps:

### Step 1: Determine the value of \( \theta \)
Since \( \tan \theta = 1 \), we know that \( \theta \) corresponds to an angle where the tangent is equal to 1. This occurs at:

\[ \theta = 45^\circ \]

### Step 2: Determine the value of \( \phi \)
Given that \( \sin \phi = \frac{1}{\sqrt{2}} \), we can recognize that this value corresponds to the sine of \( 45^\circ \):

\[ \phi = 45^\circ \]

### Step 3: Calculate \( \theta + \phi \)
Now that we have both angles, we can find \( \theta + \phi \):

\[ \theta + \phi = 45^\circ + 45^\circ = 90^\circ \]

### Step 4: Calculate \( \cos(\theta + \phi) \)
Now we need to find \( \cos(90^\circ) \):

\[ \cos(90^\circ) = 0 \]

### Final Answer
Thus, the value of \( \cos(\theta + \phi) \) is:

\[ \boxed{0} \]