Activity 15 Mach 2025 $$ \frac{\text { (1) } \sin (90+\beta) \cdot \tan (-\beta) \tan (180-\beta)}{\tan (360-\beta)} $$ $$ \frac{\cos (90+\theta) \cdot \tan (360+\theta)-\sin (360-\theta) \cdot \tan (180-\theta)}{\sin \left(\theta+180^{\circ}\right) \tan \left(\theta-180^{\circ}\right)} $$ (3) $$ \frac{\sin (180+x) \cdot \sin (-x)+\cos (40+x) \cdot \cos (x-450)}{\sin ^{2}(180+x)} $$ (4) $$ 2 \sin 150^{\circ} \cdot \cos 325^{\circ}-\sin \left(-55^{\circ}\right) $$ $$ \cos 395^{\circ} $$

Question

UpStudy AI · Accepted Answer

Let's solve each of the expressions step by step.

### Expression (1)
$$
\frac{\sin(90+\beta) \cdot \tan(-\beta) \tan(180-\beta)}{\tan(360-\beta)}
$$

1. **Calculate $$\sin(90+\beta)$$**:
   $$
   \sin(90+\beta) = \cos(\beta)
   $$

2. **Calculate $$\tan(-\beta)$$**:
   $$
   \tan(-\beta) = -\tan(\beta)
   $$

3. **Calculate $$\tan(180-\beta)$$**:
   $$
   \tan(180-\beta) = -\tan(\beta)
   $$

4. **Calculate $$\tan(360-\beta)$$**:
   $$
   \tan(360-\beta) = \tan(-\beta) = -\tan(\beta)
   $$

Now substituting these values into the expression:
$$
\frac{\cos(\beta) \cdot (-\tan(\beta)) \cdot (-\tan(\beta))}{-\tan(\beta)} = \frac{\cos(\beta) \cdot \tan^2(\beta)}{-\tan(\beta)}
$$

This simplifies to:
$$
-\cos(\beta) \cdot \tan(\beta)
$$

### Expression (2)
$$
\frac{\cos(90+\theta) \cdot \tan(360+\theta) - \sin(360-\theta) \cdot \tan(180-\theta)}{\sin(\theta+180^{\circ}) \tan(\theta-180^{\circ})}
$$

1. **Calculate $$\cos(90+\theta)$$**:
   $$
   \cos(90+\theta) = -\sin(\theta)
   $$

2. **Calculate $$\tan(360+\theta)$$**:
   $$
   \tan(360+\theta) = \tan(\theta)
   $$

3. **Calculate $$\sin(360-\theta)$$**:
   $$
   \sin(360-\theta) = -\sin(\theta)
   $$

4. **Calculate $$\tan(180-\theta)$$**:
   $$
   \tan(180-\theta) = -\tan(\theta)
   $$

5. **Calculate $$\sin(\theta+180^{\circ})$$**:
   $$
   \sin(\theta+180^{\circ}) = -\sin(\theta)
   $$

6. **Calculate $$\tan(\theta-180^{\circ})$$**:
   $$
   \tan(\theta-180^{\circ}) = \tan(\theta)
   $$

Now substituting these values into the expression:
$$
\frac{(-\sin(\theta)) \cdot \tan(\theta) - (-\sin(\theta)) \cdot (-\tan(\theta))}{(-\sin(\theta)) \cdot \tan(\theta)}
$$

This simplifies to:
$$
\frac{-\sin(\theta) \tan(\theta) + \sin(\theta) \tan(\theta)}{-\sin(\theta) \tan(\theta)} = \frac{0}{-\sin(\theta) \tan(\theta)} = 0
$$

### Expression (3)
$$
\frac{\sin(180+x) \cdot \sin(-x) + \cos(40+x) \cdot \cos(x-450)}{\sin^{2}(180+x)}
$$

1. **Calculate $$\sin(180+x)$$**:
   $$
   \sin(180+x) = -\sin(x)
   $$

2. **Calculate $$\sin(-x)$$**:
   $$
   \sin(-x) = -\sin(x)
   $$

3. **Calculate $$\sin^{2}(180+x)$$**:
   $$
   \sin^{2}(180+x) = \sin^{2}(x)
   $$

4. **Calculate $$\cos(40+x)$$** and $$\cos(x-450)$$:
   $$
   \cos(x-450) = \cos(x-90) = \sin(x)
   $$

Now substituting these values into the expression:
$$
\frac{(-\sin(x)) \cdot (-\sin(x)) + \cos(40+x) \cdot \sin(x)}{\sin^{2}(x)} = \frac{\sin^{2}(x) + \cos(40+x) \cdot \sin(x)}{\sin^{2}(x)}
$$

This simplifies to:
$$
1 + \frac{\cos(40+x)}{\sin(x)}
$$

### Expression (4)
$$
2 \sin 150^{\circ} \cdot \cos 325^{\circ} - \sin(-55^{\circ}) \cdot \cos 395^{\circ}
$$

1. **Calculate $$\sin 150^{\circ}$$**:
   $$
   \sin 150^{\circ} = \frac{1}{2}
   $$

2. **Calculate $$\cos 325^{\circ}$$**:
   $$
   \cos 325^{\circ} = \cos(360^{\circ} - 35^{\circ}) = \cos 35^{\circ}
   $$

3. **Calculate $$\sin(-55^{\circ})$$**:
   $$
   \sin(-55^{\circ}) = -\sin(55^{\circ})
   $$

4. **Calculate $$\cos 395^{\circ}$$**:
   $$
   \cos 395^{\circ} = \cos(360^{\circ} + 35^{\circ}) = \cos 35^{\circ}
   $$

Now substituting these values into the expression:
$$
2 \cdot \frac{1}{2} \cdot \cos 35^{\circ} - (-\sin(55^{\circ}) \cdot \cos 35^{\circ}) = \cos 35^{\circ} + \sin(55^{\circ}) \cdot \cos 35^{\circ}
$$

This simplifies to:
$$
\cos 35^{\circ} (1 + \sin(55^{\circ}))
$$

### Summary of Results
1. Expression (1): $$-\cos(\beta) \cdot \tan(\beta)$$
2. Expression (2): $$0$$
3. Expression (3): $$1 + \frac{\cos(40+x)}{\sin(x)}$$
4. Expression (4): $$\cos 35^{\circ} (1 + \sin(55^{\circ}))$$
Here are the simplified results for each expression: 1. **Expression (1):** $$-\cos(\beta) \cdot \tan(\beta)$$ 2. **Expression (2):** $$0$$ 3. **Expression (3):** $$1 + \frac{\cos(40+x)}{\sin(x)}$$ 4. **Expression (4):** $$\cos 35^{\circ} (1 + \sin(55^{\circ}))$$

Activity
15 Mach 2025

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(4)

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