Evaluate the expression: (sin 135° - cos 360°) / (sin 150° + cos 225°)

• For $\sin 135^\circ$: $135^\circ$ is in Quadrant II. The reference angle is $180^\circ - 135^\circ = 45^\circ$. In Quadrant II, sine is positive.

  $$\sin 135^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$$

• For $\cos 360^\circ$: $360^\circ$ is equivalent to $0^\circ$.

  $$\cos 360^\circ = \cos 0^\circ = 1$$

• For $\sin 150^\circ$: $150^\circ$ is in Quadrant II. The reference angle is $180^\circ - 150^\circ = 30^\circ$. In Quadrant II, sine is positive.

  $$\sin 150^\circ = \sin 30^\circ = \frac{1}{2}$$

• For $\cos 225^\circ$: $225^\circ$ is in Quadrant III. The reference angle is $225^\circ - 180^\circ = 45^\circ$. In Quadrant III, cosine is negative.

  $$\cos 225^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2}$$

The expression is:

Substitute the values:

Numerator:

Denominator:

Multiply the numerator and denominator by the conjugate of the denominator, which is $1 + \sqrt{2}$.

Expand the numerator:

Expand the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$:

Now, substitute these back into the fraction:

The simplified expression is $\sqrt{2}$.