How to Solve The Redundant Literal Rule (AND)? (A . B = A . (A̅ +B))
How to prove them? (Tried so many times)

Use the distributive law and you get
$$
A\cdot\bar{A} + A\cdot B.
$$
$\text{True AND False}$ results in $\text{False}$. Which give $A\cdot \bar{A} = 0$. Again, $\text{False OR X}$ always results in $X$, hence we get $0 + AB = AB$. Thus,
$$
A(\bar{A} + B) = A\cdot\bar{A} + AB = AB$$
