Answer
$(A\cap B)^{c}=A^{c}\cup B^{c} \\
to\,\,prove\,\,this\,\,we\,\,must\,\,prove\,\,\\
1-(A\cap B)^{c}\subseteq A^{c}\cup B^{c} \\
2-A^{c}\cup B^{c}\subseteq (A\cap B)^{c} \\
proof\,\,of\,\,1:\\
x\in (A\cap B)^{c}\Rightarrow x\notin(A\cap B)\\
\Rightarrow x\notin A \,or\,x\notin B \\
\Rightarrow x\in A^{c} \,\,or\,\,x\in B^{c} \\
\Rightarrow x\in A^{c}\cup B^{c}\\
\because x\in (A\cap B)^{c}\Rightarrow x\in A^{c}\cup B^{c}\\
\therefore (A\cap B)^{c}\subseteq A^{c}\cup B^{c}\\
$
$proof\,\,of\,\,2:\\
x\in A^{c}\cup B^{c} \Rightarrow x\in A^{c}\,or\,x\in B^{c}\\
\Rightarrow x\notin A \,or\,x\notin B \\
\Rightarrow x\notin (A\cap B)\\
\Rightarrow x\in (A\cap B)^{c} \\
\because x\in A^{c}\cup B^{c}\Rightarrow x\in (A\cap B)^{c}\\
\therefore A^{c}\cup B^{c}\subseteq (A\cap B)^{c}\\
from\,1\,,2\,\\
(A\cap B)^{c}=A^{c}\cup B^{c} \\
$
Work Step by Step
$(A\cap B)^{c}=A^{c}\cup B^{c} \\
to\,\,prove\,\,this\,\,we\,\,must\,\,prove\,\,\\
1-(A\cap B)^{c}\subseteq A^{c}\cup B^{c} \\
2-A^{c}\cup B^{c}\subseteq (A\cap B)^{c} \\
proof\,\,of\,\,1:\\
x\in (A\cap B)^{c}\Rightarrow x\notin(A\cap B)\\
\Rightarrow x\notin A \,or\,x\notin B \\
\Rightarrow x\in A^{c} \,\,or\,\,x\in B^{c} \\
\Rightarrow x\in A^{c}\cup B^{c}\\
\because x\in (A\cap B)^{c}\Rightarrow x\in A^{c}\cup B^{c}\\
\therefore (A\cap B)^{c}\subseteq A^{c}\cup B^{c}\\
$
$proof\,\,of\,\,2:\\
x\in A^{c}\cup B^{c} \Rightarrow x\in A^{c}\,or\,x\in B^{c}\\
\Rightarrow x\notin A \,or\,x\notin B \\
\Rightarrow x\notin (A\cap B)\\
\Rightarrow x\in (A\cap B)^{c} \\
\because x\in A^{c}\cup B^{c}\Rightarrow x\in (A\cap B)^{c}\\
\therefore A^{c}\cup B^{c}\subseteq (A\cap B)^{c}\\
from\,1\,,2\,\\
(A\cap B)^{c}=A^{c}\cup B^{c} \\
$