5.3 Simplify to a siople trigonometric ratio: \[ \sin \left(360^{\circ}+x\right) \cdot \cos \left(90^{\circ}+x\right)-\frac{\sin x}{\cos (-x) \cdot \operatorname{len}\left(360^{\circ}-x\right)} \]

To simplify the expression, let's break it down step by step. First, use the periodic properties of sine and cosine functions: \(\sin(360^{\circ} + x) = \sin x\) because sine has a period of \(360^{\circ}\), and \(\cos(90^{\circ} + x) = -\sin x\) due to the complementary angle identity. So, applying these: \[ \sin(360^{\circ} + x) \cdot \cos(90^{\circ} + x) = \sin x \cdot (-\sin x) = -\sin^2 x. \] Next, consider the second term: The cosine function is even, so \(\cos(-x) = \cos x\), and \(\operatorname{len}(360^{\circ} - x) = 360^{\circ} - x\) as it's a simple arithmetical manipulation. Normally, it just denotes a value, so we'll proceed with \(\cos(360^{\circ} - x) = \cos x\). Thus, we can rewrite the second term as: \[ -\frac{\sin x}{\cos x \cdot (360^{\circ} - x)}. \] Now combining both parts, we have: \[ -\sin^2 x + \frac{\sin x}{\cos x \cdot (360^{\circ} - x)}. \] Therefore, the simplified form becomes: \[ -\sin^2 x - \frac{\sin x}{\cos x (360^{\circ} - x)}. \] This can be expressed as: \[ -\sin^2 x - \tan x \cdot \frac{1}{(360^{\circ} - x)}. \] Thus, the overall expression simplifies to: \[ -\sin^2 x - \frac{\sin x}{\cos x(360^{\circ} - x)}. \]