S. 2 Simplify the following expeasion to a single irigatominetric tein. \[ \frac{\sin \left(360^{2}-x\right) \cdot \tan (-x)}{\cos \left(1800^{3}+x\right)\left(\sin ^{2} A+\cos ^{2} A\right)} \] S.2. Simplify fully; \( \frac{\sin 192^{\circ} \cdot \tan 300^{\circ} \cos \left(2070^{2}+x\right)}{\sin \left(-x-100^{\circ}\right) \cdot \cos 107^{\circ}} \)

To simplify the given expressions, let's tackle them step by step. 1. **First Expression Simplification**: \[ \frac{\sin (360^{2} - x) \cdot \tan (-x)}{\cos (1800^{3} + x)(\sin^{2} A + \cos^{2} A)} \] We know that \(\sin(360^{2} - x) = \sin(-x) = -\sin(x)\) since \(\sin\) is an odd function. For \(\tan(-x) = -\tan(x)\). The denominator simplifies nicely because \(\sin^{2} A + \cos^{2} A = 1\). The term \(\cos(1800^{3} + x) = \cos(x)\) because \(1800^{3}\) is a multiple of \(360^{\circ}\). Thus, our expression simplifies to: \[ \frac{-\sin(x)(-\tan(x))}{\cos(x)} = \frac{\sin(x) \tan(x)}{\cos(x)} = \sin^2(x). \] 2. **Second Expression Simplification**: \[ \frac{\sin 192^{\circ} \cdot \tan 300^{\circ} \cos (2070^{2} + x)}{\sin(-x - 100^{\circ}) \cdot \cos 107^{\circ}} \] Here, \(\tan(300^{\circ}) = \tan(360^{\circ} - 60^{\circ}) = -\tan(60^{\circ}) = -\sqrt{3}\), \(\sin(192^{\circ}) = -\sin(12^{\circ})\) (since it's in the third quadrant), and \(\sin(-x - 100^{\circ}) = -\sin(x + 100^{\circ})\). The cosine term, \(\cos(107^{\circ})\), is simply a constant value. We thus rewrite the expression while factoring in these simplifications: \[ -\frac{\sin(12^{\circ}) \cdot (-\sqrt{3}) \cdot \cos(2070^{2} + x)}{-\sin(x + 100^{\circ}) \cdot \cos(107^{\circ})} = \frac{\sin(12^{\circ}) \cdot \sqrt{3} \cdot \cos(2070^{2} + x)}{\sin(x + 100^{\circ}) \cdot \cos(107^{\circ})}. \] So, the final simplifications give us: - First Expression: \(\sin^2(x)\) - Second Expression: \(\frac{\sin(12^{\circ}) \cdot \sqrt{3} \cdot \cos(2070^{2} + x)}{\sin(x + 100^{\circ}) \cdot \cos(107^{\circ})}\)