0.4) $$ \frac{2 an (x+180) \cdot \cos (720-x)}{\cos \left(450^{\circ}+x ight)}+1 $$

Question

0.4) $$ \frac{2 	an (x+180) \cdot \cos (720-x)}{\cos \left(450^{\circ}+xight)}+1 $$

UpStudy AI · Accepted Answer

Let's simplify the given trigonometric expression step by step:

$$
\frac{2 \tan (x + 180^\circ) \cdot \cos (720^\circ - x)}{\cos (450^\circ + x)} + 1
$$

### Step 1: Simplify Each Trigonometric Function

1. **Simplify $$\tan(x + 180^\circ)$$:**
   
   The tangent function has a period of $$180^\circ$$, meaning:
   
   $$
   \tan(x + 180^\circ) = \tan(x)
   $$
   
2. **Simplify $$\cos(720^\circ - x)$$:**
   
   The cosine function has a period of $$360^\circ$$, so:
   
   $$
   \cos(720^\circ - x) = \cos(-x) = \cos(x)
   $$
   
   (Recall that $$\cos(-x) = \cos(x)$$ because cosine is an even function.)
   
3. **Simplify $$\cos(450^\circ + x)$$:**
   
   Break down $$450^\circ$$ into $$360^\circ + 90^\circ$$:
   
   $$
   \cos(450^\circ + x) = \cos(90^\circ + x)
   $$
   
   Using the cosine addition formula:
   
   $$
   \cos(90^\circ + x) = \cos(90^\circ)\cos(x) - \sin(90^\circ)\sin(x) = 0 - 1 \cdot \sin(x) = -\sin(x)
   $$

### Step 2: Substitute Simplified Functions Back into the Expression

$$
\frac{2 \tan(x) \cdot \cos(x)}{-\sin(x)} + 1
$$

### Step 3: Simplify the Fraction

1. **Express $$\tan(x)$$ in terms of $$\sin(x)$$ and $$\cos(x)$$:**
   
   $$
   \tan(x) = \frac{\sin(x)}{\cos(x)}
   $$
   
2. **Substitute $$\tan(x)$$ back into the expression:**
   
   $$
   \frac{2 \cdot \frac{\sin(x)}{\cos(x)} \cdot \cos(x)}{-\sin(x)} + 1 = \frac{2 \sin(x)}{-\sin(x)} + 1
   $$
   
3. **Simplify the fraction:**
   
   $$
   \frac{2 \sin(x)}{-\sin(x)} = -2
   $$
   
4. **Add the constant term:**
   
   $$
   -2 + 1 = -1
   $$

### Final Answer

The expression simplifies to:

$$
-1
$$

**Note:** This simplification holds for all values of $$x$$ where $$\sin(x) \neq 0$$, as division by zero is undefined.
The expression simplifies to $$-1$$.

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