$$ \left. \begin{array} { c } { A ( \beta - 180 ^ { \circ } ) an ( \beta - 180 ) \cdot \cos ( 180 + \beta } \ { \cos ( \beta ) \sin ( 200 ^ { \circ } + \beta ) } \end{array} ight. $$

Question

$$ \left. \begin{array} { c } { A ( \beta - 180 ^ { \circ } ) 	an ( \beta - 180 ) \cdot \cos ( 180 + \beta } \ { \cos ( \beta ) \sin ( 200 ^ { \circ } + \beta ) } \end{array} ight. $$

UpStudy AI · Accepted Answer

We start with the two given expressions:

$$
\begin{array}{rcl}
\text{(1)} & : & A (\beta-180^\circ) \tan (\beta-180^\circ) \cdot \cos (180^\circ+\beta)\[1mm]
\text{(2)} & : & \cos (\beta) \sin (200^\circ+\beta)
\end{array}
$$

We simplify each step by step.

---

**Expression (1):**

1. **Periodicity of the tangent function:**  
   The tangent function has a period of $$180^\circ$$. Thus,
   $$
   \tan (\beta-180^\circ)=\tan (\beta).
   $$

2. **Cosine of a shifted angle:**  
   The cosine function has the property:
   $$
   \cos(180^\circ+\beta)=-\cos (\beta).
   $$

3. **Substitute these results:**  
   Replace $$\tan (\beta-180^\circ)$$ and $$\cos (180^\circ+\beta)$$ in the expression,
   $$
   A (\beta-180^\circ) \, \tan (\beta) \cdot \bigl[-\cos (\beta)\bigr].
   $$
   This simplifies to:
   $$
   -A (\beta-180^\circ) \, \tan (\beta) \cos (\beta).
   $$

4. **Express tangent in terms of sine and cosine:**  
   Since 
   $$
   \tan (\beta)=\frac{\sin (\beta)}{\cos (\beta)},
   $$
   substitute into the expression:
   $$
   -A (\beta-180^\circ) \frac{\sin (\beta)}{\cos (\beta)} \cos (\beta).
   $$
   The factors $$\cos(\beta)$$ cancel:
   $$
   -A (\beta-180^\circ) \sin (\beta).
   $$

Thus, the simplified form of expression (1) is:
$$
-A (\beta-180^\circ) \sin (\beta).
$$

---

**Expression (2):**

1. **Rewrite the angle inside the sine function:**  
   Notice that
   $$
   200^\circ+\beta = 180^\circ + (\beta+20^\circ).
   $$

2. **Use the sine shift formula:**  
   We know that
   $$
   \sin (180^\circ+\theta)=-\sin (\theta).
   $$
   Therefore,
   $$
   \sin \bigl(200^\circ+\beta\bigr)=\sin \bigl(180^\circ+(\beta+20^\circ)\bigr)=-\sin (\beta+20^\circ).
   $$

3. **Substitute back in expression (2):**  
   $$
   \cos (\beta) \sin (200^\circ+\beta)=\cos (\beta) \bigl[-\sin (\beta+20^\circ)\bigr].
   $$
   This simplifies to:
   $$
   -\cos (\beta) \sin (\beta+20^\circ).
   $$

Thus, the simplified form of expression (2) is:
$$
-\cos (\beta) \sin (\beta+20^\circ).
$$

---

**Final Answers:**

1. $$\displaystyle -A (\beta-180^\circ) \sin (\beta)$$

2. $$\displaystyle -\cos (\beta) \sin (\beta+20^\circ)$$
The simplified forms of the expressions are: 1. $$ -A (\beta - 180^\circ) \sin (\beta) $$ 2. $$ -\cos (\beta) \sin (\beta + 20^\circ) $$

\( \left. \begin{array} { c } { A ( \beta - 180 ^ { \circ } ) \tan ( \beta - 180 ) \cdot \cos ( 180 + \beta } \\ { \cos ( \beta ) \sin ( 200 ^ { \circ } + \beta ) } \end{array} \right. \)

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