Explanation

(b)
Given, $\cos{20{^\circ} - \sin{20{^\circ} = p}}$
$\Rightarrow \cos^{2}{20{^\circ} + \sin^{2}{20{^\circ} - 2\sin{20{^\circ}\cos{20{^\circ} = p^{2}}}}}$
$\Rightarrow \ 1 - p^{2} = \sin{40{^\circ}}$
$\Rightarrow 1 - p^{2} = \sqrt{1 - \cos^{2}{40{^\circ}}}$
$\Rightarrow \left( 1 - p^{2} \right)^{2} = 1 - \cos^{2}{40{^\circ}}$
$\Rightarrow \ \cos^{2}{40{^\circ} = 1 - (1 + p^{4} - 2p^{2})}$
$\Rightarrow \cos{40{^\circ} = \sqrt{2p^{2} - p^{4}}}$
$\Rightarrow \cos{40{^\circ} = p\sqrt{2 - p^{2}}}$