Answer
See the full explanation below.
Work Step by Step
$\frac{\sin \theta -\cos \theta }{\sin \theta }+\frac{\cos \theta -\sin \theta }{\cos \theta }$
In the above expression, the least common denominator is $\left( \sin \theta \right)\left( \cos \theta \right)$. Rewrite each fraction with the least common denominator:
$\begin{align}
& \frac{\sin \theta -\cos \theta }{\sin \theta }+\frac{\cos \theta -\sin \theta }{\cos \theta }=\frac{\cos \theta \left( \sin \theta -\cos \theta \right)+\sin \theta \left( \cos \theta -\sin \theta \right)}{\sin \theta \cos \theta } \\
& =\frac{\cos \theta sin\theta -co{{s}^{2}}\theta +\sin \theta \cos \theta -{{\sin }^{2}}\theta }{\sin \theta \cos \theta } \\
& =\frac{2\sin \theta \cos \theta -\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)}{\sin \theta \cos \theta }
\end{align}$
Apply the Pythagorean identity of trigonometry ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. Then, the above expression can be further simplified as:
$\frac{2\sin \theta \cos \theta -\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)}{\sin \theta \cos \theta }=\frac{2\sin \theta \cos \theta -1}{\sin \theta \cos \theta }$
Now, the above expression can be written as:
$\frac{2\sin \theta \cos \theta -1}{\sin \theta \cos \theta }=\frac{2\sin \theta \cos \theta }{\sin \theta \cos \theta }-\frac{1}{\sin \theta \cos \theta }$
Express the terms in the numerator and denominator with the least common denominator $\sin \theta \cos \theta $:
$\frac{2\sin \theta \cos \theta }{\sin \theta \cos \theta }-\frac{1}{\sin \theta \cos \theta }=\frac{2-\sin \theta \cos \theta }{\sin \theta \cos \theta }$
Now, the above expression can be written as:
$\frac{2-\sin \theta \cos \theta }{\sin \theta \cos \theta }=2-\frac{1}{\sin \theta }.\frac{1}{\cos \theta }$
By using the reciprocal identity of trigonometry $\csc \theta =\frac{1}{sin\theta }$, and $\sec \theta =\frac{1}{\cos \theta }$, now, the above expression can be further simplified as:
$\begin{align}
& 2-\frac{1}{\sin \theta }.\frac{1}{\cos \theta }=2-\csc \theta .\sec \theta \\
& =2-\csc \theta \sec \theta
\end{align}$
Thus, the left side of the expression is equal to the right side, which is
$\frac{\sin \theta -\cos \theta }{\sin \theta }+\frac{\cos \theta -\sin \theta }{\cos \theta }=2-\sec \theta \csc \theta $.
