You can put this solution on YOUR website! Whenever a Trig. problem refers to "exact value" it is telling you that that the value can be found without calculators. And this means that "special" triangles are involved.
Let's look at a diagram of your problem. When using diagrams it is most helpful to include the unit circle (a circle of radius 1) because it makes the Trig. ratios easier to find.
For any point on the unit circle, the x-coordinate is the cos and the y-coordinate is the sin of the angle whose side passes through that point. And since the other Trig. functions can be expressed in terms of sin and cos, we can find any Trig. value with the x and y coordinates.
The side of our 135 degree angle intersects the unit circle at point A. So the x-coordinate of point A is the cos(135) and the y-coordinate is the sin(135). And since cot = cos/sin, the cot(135) = x/y. Now we just need to find the coordinates of point A!
Since our angle is 135 degrees, the angle AOB must be 45 degrees. Angle OBA is a right angle so angle OAB must be 45 degrees. That makes triangle AOB a 45-45-90 right triangle. From Geometry we know that there are certain relationships between the sides of 45-45-90 right triangles. One relationship is that the hypotenuse (OA) is always times as long as the sides opposite the 45 degree angles (OB, AB). Putting this the other way around, the sides opposite the 45 degree angles is hypotenuse divided by .
Since we are using a unit circle, our hypotenuse is 1. So that makes AB and OB . AB is the y-coordinate of point A so the sin(135) = . OB is a length so it it is positive. But point A is in the second quadrant. So its x-coordinate is negative. So the x-coordinate of A, which is the cos(135), is . The cot(135) = cos(135)/sin(135) =
(