Problem
Use the following information to answer the next question Chantal simplified the expression $\frac{\csc \theta+\sec \theta}{\sin \theta+\cos \theta}$ as shown below. \begin{tabular}{|c|c|} \hline Steps & Simplification \\ \hline 1 & $\frac{\frac{1}{\sin \theta}+\frac{1}{\cos \theta}}{\sin \theta+\cos \theta}$ \\ \hline 2 & $\frac{\frac{\cos \theta+\sin \theta}{\sin \theta}}{\sin \theta+\cos \theta}$ \\ \hline 3 & $\left(\frac{\cos \theta+\sin \theta}{\sin \theta}\right)\left(\frac{1}{\sin \theta+\cos \theta}\right)$ \\ \hline 4 & $\frac{1}{\sin \theta}$ \\ \hline 5 & $\sec \theta$ \\ \hline \end{tabular} 19. At what step did Chantal make her first mistake? 3 2 4 5
Solution
Step 1 ：The first step is correct, as \(\csc \theta\) and \(\sec \theta\) are indeed equal to \(\frac{1}{\sin \theta}\) and \(\frac{1}{\cos \theta}\) respectively.
Step 2 ：The second step is where the mistake is made. The numerator should be \(\frac{\cos \theta+\sin \theta}{\sin \theta \cos \theta}\), not \(\frac{\cos \theta+\sin \theta}{\sin \theta}\).
Step 3 ：So, the first mistake is made in step 2.
Step 4 ：Final Answer: \(\boxed{2}\)
