Explanation

(b)
$\left( \sin{x + \cos x} \right)^{2} = \frac{1}{25}$
$\Rightarrow \ \ \sin^{2}{x + \cos^{2}{x + 2\sin{x\cos{x = \frac{1}{25}}}}}$
$\Rightarrow \ \ \sin{2x = \frac{1}{25} - 1 = - \frac{24}{25}}$ …(i)
$\Rightarrow \ \cos{2x = \sqrt{1 - \sin^{2}{2x}} = - \frac{\sqrt{49}}{25}}$ …(ii)
Now, $\tan{2x = \frac{\sin{2x}}{\cos{2x}}} = - \frac{24}{25} \times \left( - \frac{25}{\sqrt{49}} \right) = \frac{24}{7}$