Answer
See the verification below.
Work Step by Step
Using the trigonometric identity
$\sin \left( 2x \right)=2\sin x\cos x$
and ${{\sin }^{2}}x+{{\cos }^{2}}x=1$
we will solve the provided identity.
$\begin{align}
& \sin \left( 2x \right)=2\sin x\cos x+\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right) \\
& ={{\sin }^{2}}x+2\sin x\cos x+{{\cos }^{2}}x
\end{align}$
Using the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$,
we will solve the provided expression
${{\left( \sin x+\cos x \right)}^{2}}$
${{\left( \sin x+\cos x \right)}^{2}}={{\sin }^{2}}x+2\sin x\cos x+{{\cos }^{2}}x$
Thus,
$\sin (2x)+1={{\left( \sin x+\cos x \right)}^{2}}$
Hence, $\sin \left( 2x \right)+1={{\left( \sin x+\cos x \right)}^{2}}$.
