Problem 13 - Precalculus Essentials

Question

Verify each identity. $$\frac{\tan \theta \cot \theta}{\csc \theta}=\sin \theta$$

Step-by-Step Solution

Answer

After simplifying \(\frac{\tan \theta \cot \theta}{\csc \theta}\), it is shown to be equivalent to \(\sin \theta\), therefore the identity is true.

Step 1: Simplify \(\tan \theta \cot \theta\)

First, let's simplify \(\tan \theta \cot \theta\). We know that \(\cot \theta\) is equal to \(\frac{1}{\tan \theta}\), so if you multiply \(\tan \theta\) by its reciprocal, you'll get 1.

Step 2: Simplify \(\frac{1}{\csc \theta}\)

Next, let's simplify \(\frac{1}{\csc \theta}\). Remember, \(\csc \theta\) is the reciprocal of \(\sin \theta\). So \(\frac{1}{\csc \theta}\) is equal to \(\sin \theta\). Now, if we rewrite the left-hand side of the equation with our simplifications, we'll get \(\sin \theta\).

Step 3: Check equivalence

The last step is to compare our simplified version of the left-hand side with the right-hand side. As our simplified expression is \(\sin \theta\), and that is exactly what we have on the right-hand side, we have successfully verified the given trigonometric identity to be true.