To rewrite the expression $\frac{\csc \theta}{\cot \theta}$ in terms of sine and cosine, follow these steps:

1. Recall the identities:

   • $\csc \theta = \frac{1}{\sin \theta}$
   • $\cot \theta = \frac{\cos \theta}{\sin \theta}$

2. Substitute these identities into the original expression:

   $\frac{\csc \theta}{\cot \theta} = \frac{\frac{1}{\sin \theta}}{\frac{\cos \theta}{\sin \theta}}$

3. Simplify the complex fraction:

   $\frac{\frac{1}{\sin \theta}}{\frac{\cos \theta}{\sin \theta}} = \frac{1}{\sin \theta} \times \frac{\sin \theta}{\cos \theta}$

4. Cancel out $\sin \theta$ from the numerator and denominator**:

   $= \frac{1}{\cos \theta}$

5. Recognize that $\frac{1}{\cos \theta}$ is the secant function:

   $= \sec \theta$

So, $\frac{\csc \theta}{\cot \theta} = \sec \theta$.

Would you like more details or have any questions?

Related questions:

1. What is the reciprocal identity for secant?
2. How do you express cotangent in terms of sine and cosine?
3. Can you derive the tangent function from sine and cosine?
4. How do you simplify $\frac{\sin \theta}{\cos \theta}$?
5. What is the inverse of the secant function?

Tip:

When simplifying trigonometric expressions, always start by recalling fundamental identities like sine, cosine, and their reciprocals.