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.
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