To complete the second step of the proof, we want to rewrite the expression \(1 - \frac{\tan \theta \cos \theta}{\csc \theta}\).
Given that:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\csc \theta = \frac{1}{\sin \theta}\)
We can substitute these definitions into the expression:
\[ 1 - \frac{\tan \theta \cos \theta}{\csc \theta} = 1 - \frac{\frac{\sin \theta}{\cos \theta} \cdot \cos \theta}{\frac{1}{\sin \theta}} \]
This simplifies to:
\[ 1 - \frac{\sin \theta}{\cos \theta} \cdot \cos \theta \cdot \sin \theta = 1 - \sin^2 \theta \]
Since we want to find a suitable expression to replace \(\frac{\tan \theta \cos \theta}{\csc \theta}\) in the proof, let's simplify that step a bit more.
Given this, we need to rewrite \(\frac{\tan \theta \cos \theta}{\csc \theta}\) directly:
\[ \frac{\tan \theta \cos \theta}{\csc \theta} = \tan \theta \cos \theta \cdot \sin \theta = \frac{\sin \theta}{\cos \theta} \cdot \cos \theta \cdot \sin \theta = \sin^2 \theta \]
Thus, the expression \(1 - \sin^2 \theta\) gives us \(\cos^2 \theta\).
Now examining the provided multiple-choice options, we want one that represents the necessary form:
- \(1 - \cos^2 \theta\) → not relevant here.
- \(1 - \sin^2 \theta \cos^2 \theta\) → not matching our requirement.
- \(1 - \frac{\sin \theta \csc \theta}{\cos \theta}\) → incorrect fraction.
- \(1 - \frac{\sin \theta}{\csc \theta}\) → simplifies to \(1 - \sin^2 \theta\).
The correct choice that leads to the necessary transformation is:
\(1 - \frac{\sin \theta}{\csc \theta}\).
This choice simplifies directly to \(1 - \sin^2 \theta\), which matches the needed step to reach \(1 - \frac{\tan \theta \cos \theta}{\csc \theta}\).
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