Write the following in terms of $$ \sin (x) $$ and $$ \cos (x) $$, and then simplify if possible. Leave your answer in terms of sines and cosines only. $$ \cot (x)-\csc (x)=\square $$

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UpStudy AI · Accepted Answer

To express $$ \cot(x) - \csc(x) $$ in terms of $$ \sin(x) $$ and $$ \cos(x) $$, follow these steps:

1. **Recall the Definitions:**
   $$
   \cot(x) = \frac{\cos(x)}{\sin(x)} \quad \text{and} \quad \csc(x) = \frac{1}{\sin(x)}
   $$

2. **Substitute the Definitions:**
   $$
   \cot(x) - \csc(x) = \frac{\cos(x)}{\sin(x)} - \frac{1}{\sin(x)}
   $$

3. **Combine the Fractions:**
   $$
   \cot(x) - \csc(x) = \frac{\cos(x) - 1}{\sin(x)}
   $$

4. **Simplify Further (Optional):**
   While $$\frac{\cos(x) - 1}{\sin(x)}$$ is already in terms of sine and cosine, it can be expressed differently by factoring:
   $$
   \frac{\cos(x) - 1}{\sin(x)} = -\frac{\sin(x)}{1 + \cos(x)}
   $$
   However, both forms are equally valid and simplified in terms of sine and cosine.

**Final Answer:**
$$
\cot(x) - \csc(x) = \frac{\cos x - 1}{\sin x}
$$
$$ \cot(x) - \csc(x) = \frac{\cos x - 1}{\sin x} $$

Write the following in terms of and , and then simplify if possible. Leave your answer in
terms of sines and cosines only.

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Write the following in terms of and , and then simplify if possible. Leave your answer in terms of sines and cosines only.