\( \therefore \frac { \tan 135 ^ { \circ } + \sin ( - 225 ^ { \circ } ) \cos 315 ^ { \circ } } { \cos 240 ^ { \circ } } = \)

To solve the expression \( \frac { \tan 135 ^ { \circ } + \sin ( - 225 ^ { \circ } ) \cos 315 ^ { \circ } } { \cos 240 ^ { \circ } } \), we start by evaluating each trigonometric function step by step. 1. **Calculate \( \tan 135^\circ \)**: \[ \tan 135^\circ = \tan(180^\circ - 45^\circ) = -\tan 45^\circ = -1 \] 2. **Calculate \( \sin(-225^\circ) \)**: \[ \sin(-225^\circ) = -\sin(225^\circ) = -\left(-\frac{1}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} \] 3. **Calculate \( \cos(315^\circ) \)**: \[ \cos(315^\circ) = \cos(360^\circ - 45^\circ) = \cos(-45^\circ) = \frac{1}{\sqrt{2}} \] 4. **Now, compute \( \sin(-225^\circ) \cos(315^\circ) \)**: \[ \sin(-225^\circ) \cos(315^\circ) = \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2} \] 5. **Next, calculate \( \cos(240^\circ) \)**: \[ \cos(240^\circ) = \cos(180^\circ + 60^\circ) = -\cos 60^\circ = -\frac{1}{2} \] 6. **Now combine everything:** \[ \tan 135^\circ + \sin(-225^\circ) \cos(315^\circ) = -1 + \frac{1}{2} = -\frac{1}{2} \] 7. **Putting it all together:** \[ \frac{\tan 135^\circ + \sin(-225^\circ) \cos(315^\circ)}{\cos(240^\circ)} = \frac{-\frac{1}{2}}{-\frac{1}{2}} = 1 \] Therefore, the final answer is: \[ \therefore \frac { \tan 135 ^ { \circ } + \sin ( - 225 ^ { \circ } ) \cos 315 ^ { \circ } } { \cos 240 ^ { \circ } } = 1 \]