If tan ` theta = 1 and sin phi = (1)/(sqrt(2)), and theta, phi in[0,pi/2]`, then the value of ` cos (theta + phi)` is

Options

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

Answer

To solve the problem, we need to find the value of \( \cos(\theta + \phi) \) given that \( \tan \theta = 1 \) and \( \sin \phi = \frac{1}{\sqrt{2}} \), with both \( \theta \) and \( \phi \) in the interval \([0, \frac{\pi}{2}]\).

### Step-by-Step Solution:

1. **Find \( \theta \)**:
- We know that \( \tan \theta = 1 \).
- The angle \( \theta \) for which \( \tan \theta = 1 \) in the interval \([0, \frac{\pi}{2}]\) is:
\[ \theta = \frac{\pi}{4} \]

2. **Find \( \phi \)**:
- We are given \( \sin \phi = \frac{1}{\sqrt{2}} \).
- The angle \( \phi \) for which \( \sin \phi = \frac{1}{\sqrt{2}} \) in the interval \([0, \frac{\pi}{2}]\) is:
\[ \phi = \frac{\pi}{4} \]

3. **Calculate \( \theta + \phi \)**:
- Now we can add the angles:
\[ \theta + \phi = \frac{\pi}{4} + \frac{\pi}{4} = \frac{2\pi}{4} = \frac{\pi}{2} \]

4. **Find \( \cos(\theta + \phi) \)**:
- We need to find \( \cos(\theta + \phi) \):
\[ \cos(\theta + \phi) = \cos\left(\frac{\pi}{2}\right) \]
- We know that:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]

### Final Answer:
Thus, the value of \( \cos(\theta + \phi) \) is:
\[ \boxed{0} \]