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

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.