Explanation

(d)
Given that, $\sin{A = \frac{1}{\sqrt{10}}}$ and $\sin{B = \frac{1}{\sqrt{5}}}$
We know that,
$\sin{(A + B) = \sin{A\cos{B + \sin{B\cos{A\ }}}}}$
$= \frac{1}{\sqrt{10}}\sqrt{1 - \frac{1}{5}} + \ \frac{1}{\sqrt{5}}\sqrt{1 - \frac{1}{10}}$
$= \frac{1}{\sqrt{10}}\sqrt{\frac{4}{5}} + \frac{1}{\sqrt{5}}\sqrt{\frac{9}{10}}$
$= \frac{1}{\sqrt{50}}(2 + 3) = \sqrt{\frac{5}{\sqrt{50}}} = \frac{1}{\sqrt{2}}$
$\Rightarrow \sin{(A + B) = \sin\frac{\pi}{4}}$
$\Rightarrow A + B = \frac{\pi}{4}$