What is the rule to eliminate (Z AND NEG Z) in this formula?

Question

Please give me a hint how I came from the left to the right side?

$$(X \land Y) \lor (Z \land \neg Z) \lor (Z \land X) \Leftrightarrow (X \land Y) \lor (Z \land X)$$

Answer 1

Note the complement laws and identity laws $$p\wedge\neg p\equiv F$$ $$p\vee F\equiv p$$ Coming to the question, $$(X \land Y) \lor (Z \land \neg Z) \lor (Z \land X) \Leftrightarrow (X \land Y) \lor (Z \land X)$$

Note that $(Z \land \neg Z)$ can be compared with $(p\wedge\neg p)\equiv F$,

so $(Z \land \neg Z)=F$

Now we have,

$$(X \land Y) \lor F \lor (Z \land X) $$and this can be compared with $p\vee F\equiv p$ which gives $(X \land Y) \lor (Z \land X) $

So, $$(X \land Y) \lor (Z \land \neg Z) \lor (Z \land X) \Leftrightarrow (X \land Y) \lor (Z \land X)$$

Answer 2

$z \land \lnot z$ is just $\bot$, "falsum", and $x \lor \bot \Leftrightarrow x$ for all $x$, as is clear from truth tables.