Answer
See the explanation below.
Work Step by Step
The expression on the left side can be expanded by using the algebraic formulas ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ and ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$. Thus, the left side can be expressed as:
$\begin{align}
& {{\left( \sin \theta +cos\theta \right)}^{2}}+{{\left( \sin \theta -cos\theta \right)}^{2}}=\left( {{\sin }^{2}}\theta +2\cos \theta .\sin \theta +{{\cos }^{2}}\theta \right)+ \\
& \left( {{\sin }^{2}}\theta -2\cos \theta .\sin \theta +{{\cos }^{2}}\theta \right) \\
& =\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)+\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)
\end{align}$
The expression can be further simplified by applying the Pythagorean identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$
$\begin{align}
& \left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)+\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)=1+1 \\
& =2
\end{align}$
Thus, the left side is equal to the right side ${{\left( \sin \theta +cos\theta \right)}^{2}}+{{\left( \sin \theta -cos\theta \right)}^{2}}=2$.
