\( \frac { \sin x ( 80 - x ) \cdot \sin ( 180 - x ) } { \cos ( 300 ^ { \circ } - x ) \cdot \sin ( 150 - x ) \cdot \cos ( 90 - x ) } \)
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To simplify the expression \( \frac { \sin x ( 80 - x ) \cdot \sin ( 180 - x ) } { \cos ( 300 ^ { \circ } - x ) \cdot \sin ( 150 - x ) \cdot \cos ( 90 - x ) } \), we start by simplifying the trigonometric functions involved. 1. **Sin and Cos Functions:** We know that \( \sin(180 - x) = \sin x \) and \( \cos(90 - x) = \sin x \). Using these properties: \[ \sin(180 - x) = \sin x \] Replacing \( \sin(180 - x) \) in the numerator gives: \[ \sin x (80 - x) \cdot \sin x = \sin^2 x (80 - x) \] The numerator simplifies to: \[ \sin^2 x (80 - x) \] 2. **Cosine Simplifications:** We know that: \[ \cos(300^\circ - x) = \cos(360^\circ - 60^\circ - x) = \cos(-60^\circ - x) = \cos(60^\circ + x) \] We can further expand that if needed or leave it as is for the expression. The term \(\sin(150 - x)\) simplifies using \( \sin(150^\circ - x) = \sin(30^\circ + x) = \sin(30^\circ)\cos(x) + \cos(30^\circ)\sin(x) = \frac{1}{2}\cos(x) + \frac{\sqrt{3}}{2}\sin(x)\). The denominator thus becomes: \[ \cos(300^\circ - x) \cdot \left( \frac{1}{2}\cos(x) + \frac{\sqrt{3}}{2}\sin(x) \right) \cdot \sin x \] 3. **Combining it all together:** The full expression now looks like: \[ \frac {\sin^2 x (80 - x)} {\cos(300^\circ - x) \cdot \sin(150 - x) \cdot \sin x} \] Canceling \( \sin x \) from numerator and denominator gives: \[ \frac {\sin x (80 - x)} {\cos(300^\circ - x) \cdot \sin(150 - x)} \] At this point, we can leave the expression simplified as: \[ \frac{\sin x (80 - x)}{\cos(300^\circ - x) \cdot \sin(150 - x)} \] This is a cleaner representation of the original expression, using fundamental trigonometric identities to achieve the simplification.
