Questions: Rewrite csc θ / cot θ in terms of sine and cosine. csc θ / cot θ = (Simplify your answer.)

Transcript text: 1. Rewrite $\frac{\boldsymbol{\operatorname { c s c }} \theta}{\boldsymbol{\operatorname { c o t }} \theta}$ in terms of sine and cosine. \[ \frac{\boldsymbol{\operatorname { c s c }} \theta}{\boldsymbol{\operatorname { c o t }} \theta}=\square \] (Simplify your answer.)

Solution

Solution Steps

Step 1: Rewrite $\csc \theta$ and $\cot \theta$ in terms of sine and cosine

\[ \csc \theta = \frac{1}{\sin \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]

Step 2: Substitute the expressions into the original equation

\[ \frac{\csc \theta}{\cot \theta} = \frac{\frac{1}{\sin \theta}}{\frac{\cos \theta}{\sin \theta}} \]

Step 3: Simplify the fraction

\[ \frac{\frac{1}{\sin \theta}}{\frac{\cos \theta}{\sin \theta}} = \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{1}{\cos \theta} \]

Step 4: Rewrite $\frac{1}{\cos \theta}$ as $\sec \theta$

\[ \frac{1}{\cos \theta} = \sec \theta \]