The problem in the image asks to "Write the following in terms of $\sin \theta$ and $\cos \theta$; then simplify if possible":

$\sin \theta \cot \theta + 4 \cos \theta$

Let's break this down:

Step 1: Express $\cot \theta$ in terms of $\sin \theta$ and $\cos \theta$.

We know that: $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Step 2: Substitute this into the original expression.

$\sin \theta \cdot \cot \theta + 4 \cos \theta = \sin \theta \cdot \frac{\cos \theta}{\sin \theta} + 4 \cos \theta$

Step 3: Simplify the expression.

The $\sin \theta$ in the numerator and denominator cancels out in the first term: $\cos \theta + 4 \cos \theta$

Step 4: Combine like terms.

$\cos \theta + 4 \cos \theta = 5 \cos \theta$

Thus, the simplified expression is: $5 \cos \theta$

Would you like more details or explanations on any part of this process?

Relative Questions:

1. What is the cotangent function and how is it related to sine and cosine?
2. How do we simplify trigonometric expressions involving multiple functions?
3. Can we always cancel terms in trigonometric identities like we did with $\sin \theta$?
4. What happens if $\sin \theta = 0$ in this particular problem?
5. How do we apply trigonometric identities in more complex equations?

Tip:

Always express trigonometric functions in terms of sine and cosine when simplifying, as this often leads to easier cancellation and simplification.