\[(a)\;\frac{-1}{\sqrt 2}(\hat i+\hat j) \quad (b)\;\frac{1}{\sqrt 2}(\hat i+\hat j) \quad (c)\;\bigg(\frac{1}{2}\hat i+\frac{\sqrt 3}{2}\hat j\bigg) \quad (d)\;\bigg(\frac{1}{2}\hat i-\frac{\sqrt 3}{2}\hat j\bigg)\]

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Let U be the potential energy

$U(x,y)=\cos (x +y)$

$\therefore F_x=\large\frac{-\partial u}{\partial x}$

$\quad =\large\frac{-\partial }{\partial x}$$ \cos (x+y)$

$\quad=\sin (x+y)$

at $(0,\large\frac{\pi}{4})\quad$$ F_x=\large\frac{1}{\sqrt 2}$

$F_y=\large\frac{-\partial u}{\partial y}$

$\quad=\sin (x+y)$

at $(0, \large\frac{\pi}{4}) \quad $$F_y=\large\frac{1}{\sqrt 2}$

$\therefore F= F_x \hat i+F_y \hat j$

$\quad= \large\frac{1}{\sqrt 2}$$(\hat i+\hat j)$

Hence b is the correct answer.

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