Consider the following for the two (02) items that follow: Let p =...

p = |\sin\alpha - \sin(\alpha - 90^{}^{\circ})|

Using the trigonometric identity \sin(\alpha - 90^{}^{\circ}) = -\sin(90^{}^{\circ} - \alpha) = -\cos\alpha, we get:

$p = |\sin\alpha - (-\cos\alpha)| = |\sin\alpha + \cos\alpha|$

Multiplying and dividing the expression inside the modulus by $\sqrt{2}:<br><br>$p = $\sqrt{2}\left| \dfrac{1}{\sqrt{2}}\sin\alpha + \dfrac{1}{\sqrt{2}}\cos\alpha \right|$

p = \sqrt{2} |\sin(\alpha + 45^{}^{\circ})|

Since the absolute value of any real expression is non-negative, the minimum possible value of $p$ is $0$. This minimum is achieved when \sin(\alpha + 45^{}^{\circ}) = 0, which occurs at \alpha = -45^{}^{\circ}.

Therefore, the minimum value of $p$ is $0$.