To simplify the given expression:
[tex]\[ \frac{\tan (180^\circ - x) \cdot \cos (360^\circ - x) + \sin (540^\circ + x)}{\cos (90^\circ - x) \cdot \cos (-x)} \][/tex]
we will use trigonometric identities and properties of the trigonometric functions. Let's break this down step-by-step.
### Step 1: Simplify each component using trigonometric identities
1. Simplify [tex]\(\tan(180^\circ - x)\)[/tex]:
[tex]\[ \tan(180^\circ - x) = -\tan(x) \][/tex]
This is because the tangent function is periodic with a period of [tex]\(180^\circ\)[/tex] and changes sign in the second quadrant.
2. Simplify [tex]\(\cos(360^\circ - x)\)[/tex]:
[tex]\[ \cos(360^\circ - x) = \cos(x) \][/tex]
Since cosine is a periodic function with a period of [tex]\(360^\circ\)[/tex], [tex]\(\cos(360^\circ - x) = \cos(x)\)[/tex].
3. Simplify [tex]\(\sin(540^\circ + x)\)[/tex]:
[tex]\[ 540^\circ = 1.5 \times 360^\circ = 360^\circ + 180^\circ \][/tex]
Hence, [tex]\(\sin(540^\circ + x) = \sin(360^\circ + 180^\circ + x) = \sin(180^\circ + x) = -\sin(x)\)[/tex].
4. Simplify [tex]\(\cos(90^\circ - x)\)[/tex]:
[tex]\[ \cos(90^\circ - x) = \sin(x) \][/tex]
This follows from the co-function identity of cosine and sine.
5. Simplify [tex]\(\cos(-x)\)[/tex]:
[tex]\[ \cos(-x) = \cos(x) \][/tex]
Since cosine is an even function, it remains unchanged under a sign flip.
### Step 2: Substitute the simplified components back into the expression
Let's substitute these simplified components into the expression:
[tex]\[ \frac{\tan(180^\circ - x) \cdot \cos(360^\circ - x) + \sin(540^\circ + x)}{\cos(90^\circ - x) \cdot \cos(-x)} \][/tex]
Substituting [tex]\( \tan(180^\circ - x) = -\tan(x) \)[/tex], [tex]\( \cos(360^\circ - x) = \cos(x) \)[/tex], [tex]\( \sin(540^\circ + x) = -\sin(x) \)[/tex]:
[tex]\[ \frac{(-\tan(x)) \cdot \cos(x) + (-\sin(x))}{\sin(x) \cdot \cos(x)} \][/tex]
### Step 3: Further Simplify the Expression
1. Simplify the numerator:
[tex]\[ (-\tan(x)) \cdot \cos(x) + (-\sin(x)) = -\tan(x) \cos(x) - \sin(x) = -\left( \frac{\sin(x)}{\cos(x)} \cdot \cos(x) \right) - \sin(x) \][/tex]
Simplify the term [tex]\(\tan(x) \cos(x)\)[/tex]:
[tex]\[ -\left( \frac{\sin(x)}{\cos(x)} \cdot \cos(x) \right) = -\sin(x) \][/tex]
Thus, the numerator is:
[tex]\[ -\sin(x) - \sin(x) = -2\sin(x) \][/tex]
2. Simplify the denominator:
[tex]\[ \sin(x) \cos(x) \][/tex]
So our expression now is:
[tex]\[ \frac{-2\sin(x)}{\sin(x) \cos(x)} \][/tex]
[tex]\[ = \frac{-2\sin(x)}{\sin(x) \cos(x)} \][/tex]
[tex]\[ = \frac{-2}{\cos(x)} \][/tex]
(Since [tex]\(\sin(x)\)[/tex] in numerator and denominator cancels out)
So, the fully simplified expression is:
[tex]\[ -\frac{2}{\cos(x)} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{b. } -\frac{2}{\cos(x)}} \][/tex]
