### Instruction: Compute $\cos 135^\circ$.### Response: I know that $\cos 135^\circ$ is the same as $\cos (\pi/4 + \pi/2)$, because $\pi/4$ is 45 degrees and $\pi/2$ is 90 degrees, and adding them gives 135 degrees.
I also know that there is a trigonometric identity that says $\cos (A + B) = \cos A \cos B - \sin A \sin B$, where $A$ and $B$ are any angles.
I can use this identity to expand $\cos (\pi/4 + \pi/2)$.
Substituting $A = \pi/4$ and $B = \pi/2$, I get $\cos (\pi/4 + \pi/2) = \cos \pi/4 \cos \pi/2 - \sin \pi/4 \sin \pi/2$.
Now I need to recall the values of the cosine and sine functions for these special angles.
I can use the unit circle or a right triangle to help me remember them.
The cosine of $\pi/4$ is $\sqrt{2}/2$, because it is the x-coordinate of the point on the unit circle that makes a 45 degree angle with the positive x-axis.
Similarly, the sine of $\pi/4$ is also $\sqrt{2}/2$, because it is the y-coordinate of the same point.
The cosine of $\pi/2$ is 0, because it is the x-coordinate of the point on the unit circle that makes a 90 degree angle with the positive x-axis.
The sine of $\pi/2$ is 1, because it is the y-coordinate of the same point.