\( \frac { 4 \cos ( - x ) \cdot \cos ( 90 ^ { \circ } + x ) } { \sin ( 30 ^ { \circ } - 1 ) \cdot \cos x + \cos ( 30 ^ { \circ } - x ) \cdot \sin x } \)

Let's simplify the given expression step by step. Firstly, recall some trigonometric identities: - \( \cos(-x) = \cos(x) \) - \( \cos(90^\circ + x) = -\sin(x) \) - \( \sin(30^\circ) = \frac{1}{2} \) - \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \) Now, substituting these into the expression: 1. The numerator simplifies as follows: \[ 4 \cos(-x) \cdot \cos(90^\circ + x) = 4 \cos(x) \cdot (-\sin(x)) = -4 \cos(x) \sin(x) \] 2. The denominator requires more attention: - For the first part of the denominator: \[ \sin(30^\circ - 1) \cdot \cos(x) = \sin(-\frac{1}{2}) \cdot \cos(x) = -\sin(1) \cdot \cos(x) \] - For the second part: \[ \cos(30^\circ - x) \cdot \sin(x) = \left(\cos 30^\circ \cos x + \sin 30^\circ \sin x \right) \sin(x) = \left(\frac{\sqrt{3}}{2} \cos x + \frac{1}{2} \sin x\right) \sin(x) \] So, we can combine both parts: \[ -\sin(1) \cos(x) + \left(\frac{\sqrt{3}}{2} \cos x \sin x + \frac{1}{2} \sin^2 x\right) \] Now we can express the entire fraction: \[ \frac{-4 \cos(x) \sin(x)}{-\sin(1) \cos(x) + \left(\frac{\sqrt{3}}{2} \cos x \sin x + \frac{1}{2} \sin^2 x\right)} \] Next, we will simplify the expression further. Let's check if we can factor the denominator. We will analyze the common terms in the denominator more closely and potentially simplify major components: - Look for any common factors or standard forms within the terms to reduce them effectively. The overall evaluation and proper simplifications can yield a clearer result, depending on additional context or requirements. Ultimately, this is a non-trivial expression and each trigonometric function plays a crucial role in the outcome. Calculating precise values at specific coordinates may also yield numerical insights!