Answer
The expression on the left-side is equal to the expression on the right-side.
Work Step by Step
The given expression on the left side $\frac{\sin 2\theta -\sin \theta }{\cos 2\theta +\cos \theta }$ can be simplified by applying the double angle formula $\sin 2\theta =2\sin \theta \cos \theta $ and $\cos 2\theta =2{{\cos }^{2}}\theta -1$.
$\begin{align}
& \frac{\sin 2\theta -\sin \theta }{\cos 2\theta +\cos \theta }=\frac{2\sin \theta \cos \theta -\sin \theta }{2{{\cos }^{2}}\theta -1+\cos \theta } \\
& =\frac{\sin \theta \left( 2\cos \theta -1 \right)}{\left( 2\cos \theta -1 \right)\left( \cos \theta +1 \right)} \\
& =\frac{\sin \theta }{\cos \theta +1}
\end{align}$
Rationalizing the expression gives:
$\begin{align}
& \frac{\sin \theta }{\cos \theta +1}=\frac{\sin \theta }{\cos \theta +1}.\frac{\cos \theta -1}{\cos \theta -1} \\
& =\frac{\sin \theta \left( \cos \theta -1 \right)}{{{\cos }^{2}}\theta -1} \\
& =\frac{\sin \theta \left( \cos \theta -1 \right)}{-{{\sin }^{2}}\theta } \\
& =\frac{-\left( \cos \theta -1 \right)}{\sin \theta }
\end{align}$
Now, the expression can be written as:
$\frac{-\left( \cos \theta -1 \right)}{\sin \theta }=\frac{1-\cos \theta }{\sin \theta }$
Hence, the expression on the left-side is equal to the expression on the right-side.
