Q:

I am to ask to show that $P\cap (P\cap Q )^{c} = P\cap Q^{c}$

As a reference, here's a picture for P ∩ Q' 

A:

$\begin{align}P\cap (P\cap Q )^{c} &=P\cap( P^c\cup Q^{c})\\&=(P\cap P^c)\cup (P\cap Q^{c})\\&=(P\cap Q^{c})\end{align}$
since $P\cap P^c=\phi$.

