1. The problem asks us to evaluate the expression $\sin(135^\circ) - \cos(45^\circ)$.\n\n2. Recall the values of sine and cosine for special angles: \n- $\sin(135^\circ) = \sin(180^\circ - 45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}$ because sine is positive in the second quadrant.\n- $\cos(45^\circ) = \frac{\sqrt{2}}{2}$.\n\n3. Substitute these values into the expression:\n$$\sin(135^\circ) - \cos(45^\circ) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}$$\n\n4. Simplify the expression by subtracting the two equal terms:\n$$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0$$\n\n5. Therefore, the value of $\sin(135^\circ) - \cos(45^\circ)$ is $0$.