If $ \tan \theta=1 $ and $ \sin \phi =\frac{1}{\sqrt{2}}, $ then the value of $ \cos (\theta+\phi) $ is:
(a) -1
(b) 0
(c) 1
(d) $ \frac{\sqrt{3}}{2} $

Given:

\( \tan \theta=1 \) and \( \sin \phi =\frac{1}{\sqrt{2}} \).
To do:

We have to find the value of \( \cos (\theta+\phi) \).
Solution:
 We know that,

$tan\ 45^o=1$ and $sin\ 45^o=\frac{1}{\sqrt{2}}$

\( \tan \theta=1 \)

This implies,

$\tan \theta=tan\ 45^o$

$\theta = 45^o$

\( \sin \phi =\frac{1}{\sqrt{2}} \)

$\sin \phi =sin\ 45^o$

$\phi = 45^o$

Therefore,

$cos\ (\theta+\phi) = cos\ (45^o+45^o)$

$=cos\ 90^o$

$=0$

Option (b) is the correct answer.