3.163. $\sin 2 \alpha$, if $\sin \alpha - \cos \alpha = p$.
Solution.

$\sin \alpha-\cos \alpha=p \quad \Rightarrow$

$\Rightarrow \sin ^{2} \alpha-2 \sin \alpha \cos \alpha+\cos ^{2} \alpha=p^{2}, \quad 1-\sin 2 \alpha=p^{2}$,

from which $\sin 2 \alpha=1-p^{2}$.

Answer: $1-p^{2}$.