\( \frac { \sin ( 180 - \theta ) \tan ( 180 + \theta ) \cos ( \theta + 860 ) } { \cos ( 90 - \theta ) \cos ( - \theta ) \tan ( 360 - \theta ) } \)

To simplify the expression: \[ \frac{ \sin (180^\circ - \theta) \tan (180^\circ + \theta) \cos (\theta + 860^\circ) }{ \cos (90^\circ - \theta) \cos (-\theta) \tan (360^\circ - \theta) } \] Let's break it down step by step using trigonometric identities: 1. **Simplify Each Trigonometric Function:** - **\(\sin(180^\circ - \theta)\):** \[ \sin(180^\circ - \theta) = \sin \theta \] - **\(\tan(180^\circ + \theta)\):** \[ \tan(180^\circ + \theta) = \tan \theta \] - **\(\cos(\theta + 860^\circ)\):** \[ 860^\circ \mod 360^\circ = 140^\circ \quad \Rightarrow \quad \cos(\theta + 860^\circ) = \cos(\theta + 140^\circ) \] - **\(\cos(90^\circ - \theta)\):** \[ \cos(90^\circ - \theta) = \sin \theta \] - **\(\cos(-\theta)\):** \[ \cos(-\theta) = \cos \theta \] - **\(\tan(360^\circ - \theta)\):** \[ \tan(360^\circ - \theta) = \tan(-\theta) = -\tan \theta \] 2. **Substitute These Simplifications Back Into the Expression:** \[ \frac{ \sin \theta \cdot \tan \theta \cdot \cos(\theta + 140^\circ) }{ \sin \theta \cdot \cos \theta \cdot (-\tan \theta) } \] 3. **Cancel Out Common Terms:** - \(\sin \theta\) cancels out. - \(\tan \theta\) cancels out (assuming \(\tan \theta \neq 0\)). The expression simplifies to: \[ \frac{ \cos(\theta + 140^\circ) }{ -\cos \theta } = -\frac{ \cos(\theta + 140^\circ) }{ \cos \theta } \] **Final Simplified Form:** \[ -\frac{ \cos(\theta + 140^\circ) }{ \cos \theta } \] **Alternatively,** this can be written as: \[ - \frac{\cos(\theta + 140^\circ)}{\cos \theta} \] **Conclusion:** The simplified form of the given expression is: \[ - \frac{\cos(\theta + 140^\circ)}{\cos \theta} \] Answer: After simplifying, the expression is –cos ( θ + 140° ) divided by cos θ. Thus, − cos(θ+140°)/cosθ