Given,

$(p\wedge (~q\left)\right)\vee \left(\right(~p) \wedge q) \vee \left(\right(~p) \wedge (~q\left)\right)$

$=\left(\right(~p) \wedge q) \vee \left(\right(~p) \wedge (~q\left)\right)\vee (p\wedge (~q\left)\right)$

$=\left(~p\wedge \left(q\vee ~q\right)\right)\vee \left(p\wedge ~q\right)$

We know that $~A\vee A=T$, where $T$ is tautology,

$=\left(~p\wedge T\right)\vee \left(p\wedge ~q\right)$

$=~p\vee \left(p\wedge ~q\right)$

$=\left(~p\vee p\right)\wedge \left(~p\vee ~q\right)$

$=T\wedge \left(~p\vee ~q\right)$

$=~p\vee ~q$