1. **State the problem:** Simplify the expression $$\frac{(\cos 540)(\csc 390) + \cot (-480)}{(\cos (-45)) (\sec 480)}$$. 2. **Recall angle reduction and trigonometric identities:** - Angles can be reduced by subtracting multiples of 360° because trigonometric functions are periodic with period 360°. - $\cos(-\theta) = \cos \theta$, $\sec \theta = \frac{1}{\cos \theta}$, $\csc \theta = \frac{1}{\sin \theta}$, and $\cot \theta = \frac{\cos \theta}{\sin \theta}$. 3. **Reduce angles:** - $540^\circ = 540 - 360 = 180^\circ$ - $390^\circ = 390 - 360 = 30^\circ$ - $-480^\circ = -480 + 360 + 360 = 240^\circ$ - $-45^\circ = 360 - 45 = 315^\circ$ (or use $\cos(-45) = \cos 45$) - $480^\circ = 480 - 360 = 120^\circ$ 4. **Rewrite the expression with reduced angles:** $$\frac{(\cos 180)(\csc 30) + \cot 240}{(\cos 45)(\sec 120)}$$ 5. **Evaluate each trigonometric function:** - $\cos 180 = -1$ - $\sin 30 = \frac{1}{2} \Rightarrow \csc 30 = \frac{1}{\sin 30} = 2$ - $\cot 240 = \frac{\cos 240}{\sin 240}$ - $\cos 240 = \cos(180 + 60) = -\cos 60 = -\frac{1}{2}$ - $\sin 240 = \sin(180 + 60) = -\sin 60 = -\frac{\sqrt{3}}{2}$ - So, $\cot 240 = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}$ - $\cos 45 = \frac{\sqrt{2}}{2}$ - $\cos 120 = \cos(180 - 60) = -\cos 60 = -\frac{1}{2}$ - $\sec 120 = \frac{1}{\cos 120} = \frac{1}{-\frac{1}{2}} = -2$ 6. **Substitute values back:** $$\frac{(-1)(2) + \frac{1}{\sqrt{3}}}{\left(\frac{\sqrt{2}}{2}\right)(-2)} = \frac{-2 + \frac{1}{\sqrt{3}}}{-\sqrt{2}}$$ 7. **Simplify numerator:** $$-2 + \frac{1}{\sqrt{3}} = \frac{-2\sqrt{3} + 1}{\sqrt{3}}$$ 8. **Rewrite the whole expression:** $$\frac{\frac{-2\sqrt{3} + 1}{\sqrt{3}}}{-\sqrt{2}} = \frac{-2\sqrt{3} + 1}{\sqrt{3}} \times \frac{1}{-\sqrt{2}} = \frac{-2\sqrt{3} + 1}{\sqrt{3}} \times \frac{-1}{\sqrt{2}}$$ 9. **Multiply numerator and denominator:** $$= \frac{(-2\sqrt{3} + 1)(-1)}{\sqrt{3} \sqrt{2}} = \frac{2\sqrt{3} - 1}{\sqrt{6}}$$ 10. **Rationalize the denominator:** $$\frac{2\sqrt{3} - 1}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{(2\sqrt{3} - 1)\sqrt{6}}{6}$$ 11. **Simplify numerator:** $$ (2\sqrt{3})\sqrt{6} - 1 \times \sqrt{6} = 2\sqrt{18} - \sqrt{6} = 2 \times 3\sqrt{2} - \sqrt{6} = 6\sqrt{2} - \sqrt{6}$$ 12. **Final simplified expression:** $$\frac{6\sqrt{2} - \sqrt{6}}{6}$$ **Answer:** $$\boxed{\frac{6\sqrt{2} - \sqrt{6}}{6}}$$