1. $$ \sin x \cos y-\cos \alpha \sin y $$ $$ 2 \cdot \operatorname{Cos} a \cdot \cos 12^{\circ}+\sin a \sin ^{\circ} 12^{\circ} $$ $$ 3 \operatorname{Cos} heta \operatorname{Cos} 4 heta-\operatorname{Sin} heta \operatorname{Sin} 4 heta $$ $$ +\sin 43^{\circ} \cos 33^{\circ}-\cos 43^{\circ} \sin 33^{\circ} $$ $$ =\operatorname{Cos} 14^{\circ} \sin 6^{\circ}+\operatorname{Cos} 6^{\circ} \operatorname{Sin} A^{\circ} $$ - $$ \operatorname{Sin} 6 x \sin 5 x+\operatorname{Cos} 6 \operatorname{Cos} 5 x $$ 1. $$ \sin \left(x-20^{\circ} ight)\left(x \operatorname{ses} 20^{\circ}+\cos \left(x-20^{\circ} ight) \sin 20 ight. $$

Question

1. $$ \sin x \cos y-\cos \alpha \sin y $$ $$ 2 \cdot \operatorname{Cos} a \cdot \cos 12^{\circ}+\sin a \sin ^{\circ} 12^{\circ} $$ $$ 3 \operatorname{Cos} 	heta \operatorname{Cos} 4 	heta-\operatorname{Sin} 	heta \operatorname{Sin} 4 	heta $$ $$ +\sin 43^{\circ} \cos 33^{\circ}-\cos 43^{\circ} \sin 33^{\circ} $$ $$ =\operatorname{Cos} 14^{\circ} \sin 6^{\circ}+\operatorname{Cos} 6^{\circ} \operatorname{Sin} A^{\circ} $$ - $$ \operatorname{Sin} 6 x \sin 5 x+\operatorname{Cos} 6 \operatorname{Cos} 5 x $$ 1. $$ \sin \left(x-20^{\circ}ight)\left(x \operatorname{ses} 20^{\circ}+\cos \left(x-20^{\circ}ight) \sin 20ight. $$

UpStudy AI · Accepted Answer

Let's break down the problem step by step. The problem consists of several trigonometric expressions and equations. We will analyze each part and simplify or solve them as needed.

### Step 1: Analyze the Expressions

1. **Expression 1:** 
   $$
   \sin x \cos y - \cos \alpha \sin y
   $$

2. **Expression 2:** 
   $$
   2 \cdot \cos a \cdot \cos 12^{\circ} + \sin a \sin 12^{\circ}
   $$

3. **Expression 3:** 
   $$
   \cos \theta \cos 4\theta - \sin \theta \sin 4\theta
   $$

4. **Expression 4:** 
   $$
   \sin 43^{\circ} \cos 33^{\circ} - \cos 43^{\circ} \sin 33^{\circ}
   $$

5. **Expression 5:** 
   $$
   \cos 14^{\circ} \sin 6^{\circ} + \cos 6^{\circ} \sin A^{\circ}
   $$

6. **Expression 6:** 
   $$
   \sin 6x \sin 5x + \cos 6 \cos 5x
   $$

7. **Expression 7:** 
   $$
   \sin(x - 20^{\circ}) \left( x \cos 20^{\circ} + \cos(x - 20^{\circ}) \sin 20^{\circ} \right)
   $$

### Step 2: Simplify Each Expression

#### Expression 1:
This expression can be rewritten using the sine subtraction formula:
$$
\sin x \cos y - \cos \alpha \sin y = \sin(x - y) \text{ (if } y = \alpha \text{)}
$$

#### Expression 2:
Using the cosine addition formula:
$$
2 \cos a \cos 12^{\circ} + \sin a \sin 12^{\circ} = \cos(a - 12^{\circ})
$$

#### Expression 3:
Using the cosine subtraction formula:
$$
\cos \theta \cos 4\theta - \sin \theta \sin 4\theta = \cos(\theta + 4\theta) = \cos(5\theta)
$$

#### Expression 4:
Using the sine subtraction formula:
$$
\sin 43^{\circ} \cos 33^{\circ} - \cos 43^{\circ} \sin 33^{\circ} = \sin(43^{\circ} - 33^{\circ}) = \sin(10^{\circ})
$$

#### Expression 5:
Using the sine addition formula:
$$
\cos 14^{\circ} \sin 6^{\circ} + \cos 6^{\circ} \sin A^{\circ} = \sin(14^{\circ} + 6^{\circ}) = \sin(20^{\circ})
$$

#### Expression 6:
Using the product-to-sum identities:
$$
\sin 6x \sin 5x + \cos 6 \cos 5x = \frac{1}{2} [\cos(6x - 5x) - \cos(6x + 5x)] = \frac{1}{2} [\cos(x) - \cos(11x)]
$$

#### Expression 7:
This expression can be simplified using the sine addition formula:
$$
\sin(x - 20^{\circ}) \left( x \cos 20^{\circ} + \cos(x - 20^{\circ}) \sin 20^{\circ} \right)
$$

### Step 3: Combine and Solve

Now we can combine the simplified expressions and solve for any unknowns if necessary. However, the problem does not specify any equations to solve, so we will summarize the simplified forms:

1. $$ \sin(x - y) $$
2. $$ \cos(a - 12^{\circ}) $$
3. $$ \cos(5\theta) $$
4. $$ \sin(10^{\circ}) $$
5. $$ \sin(20^{\circ}) $$
6. $$ \frac{1}{2} [\cos(x) - \cos(11x)] $$
7. $$ \sin(x - 20^{\circ}) \left( x \cos 20^{\circ} + \cos(x - 20^{\circ}) \sin 20^{\circ} \right) $$

If you have specific values for $$x$$, $$y$$, $$a$$, $$\alpha$$, or $$\theta$$, please provide them for further calculations or evaluations.
Here are the simplified forms of the trigonometric expressions: 1. $$ \sin(x - y) $$ 2. $$ \cos(a - 12^{\circ}) $$ 3. $$ \cos(5\theta) $$ 4. $$ \sin(10^{\circ}) $$ 5. $$ \sin(20^{\circ}) $$ 6. $$ \frac{1}{2} [\cos(x) - \cos(11x)] $$ 7. $$ \sin(x - 20^{\circ}) \left( x \cos 20^{\circ} + \cos(x - 20^{\circ}) \sin 20^{\circ} \right) $$

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