1. **State the problem:** Find equivalent ratios in terms of the acute angle and then find the exact value of $\tan 150^\circ - \sin 135^\circ$.\n\n2. **Recall angle relationships:** Both $150^\circ$ and $135^\circ$ are in the second quadrant. The reference (acute) angle for $150^\circ$ is $180^\circ - 150^\circ = 30^\circ$. The reference angle for $135^\circ$ is $180^\circ - 135^\circ = 45^\circ$.\n\n3. **Use the formulas for tangent and sine in the second quadrant:**\n- $\tan(180^\circ - \theta) = -\tan \theta$\n- $\sin(180^\circ - \theta) = \sin \theta$\n\n4. **Rewrite the expressions:**\n$$\tan 150^\circ = \tan(180^\circ - 30^\circ) = -\tan 30^\circ$$\n$$\sin 135^\circ = \sin(180^\circ - 45^\circ) = \sin 45^\circ$$\n\n5. **Recall exact values:**\n$$\tan 30^\circ = \frac{1}{\sqrt{3}}$$\n$$\sin 45^\circ = \frac{\sqrt{2}}{2}$$\n\n6. **Substitute values:**\n$$\tan 150^\circ - \sin 135^\circ = -\frac{1}{\sqrt{3}} - \frac{\sqrt{2}}{2}$$\n\n7. **Rationalize the first term:**\n$$-\frac{1}{\sqrt{3}} = -\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$$\n\n8. **Final exact value:**\n$$-\frac{\sqrt{3}}{3} - \frac{\sqrt{2}}{2}$$\n\nThis is the exact value of the expression.