Answer
See the explanation below.
Work Step by Step
${{\left( \cos \theta -\sin \theta \right)}^{2}}+{{\left( \cos \theta +\sin \theta \right)}^{2}}$
Now, factorize the above expression. By using the formulae ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ and ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ , with $A=\cos \theta $ and $B=\sin \theta $ for the numeric expression.
$\begin{align}
& {{\left( \cos \theta -\sin \theta \right)}^{2}}+{{\left( \cos \theta +\sin \theta \right)}^{2}}={{\cos }^{2}}\theta -2\cos \theta \sin \theta +{{\sin }^{2}}\theta +{{\cos }^{2}}\theta +2\cos \theta \sin \theta +{{\sin }^{2}}\theta \\
& ={{\cos }^{2}}\theta +{{\sin }^{2}}\theta +{{\cos }^{2}}\theta +{{\sin }^{2}}\theta
\end{align}$
Apply the Pythagorean identity of trigonometry ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$; then the above expression can be further simplified as:
$\begin{align}
& {{\cos }^{2}}\theta +{{\sin }^{2}}\theta +{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1+1 \\
& =2
\end{align}$
Thus, the left side of the expression is equal to the right side, which is
${{\left( \cos \theta -\sin \theta \right)}^{2}}+{{\left( \cos \theta +\sin \theta \right)}^{2}}=2$.
