Find the exact value of the expression. do not use a calculator. if tan θ = 9, find the exact value of cos (θ).

Form a right triangle with angle θ. If tan(θ) = 9, we can assign the length of the side opposite the angle to be 9, and the length of the side adjacent to the angle to be 1. Then tan(θ) is the ratio between the opposite side and the adjacent side = 9.

Calculate the length of the hypotenuse to be: √(1^2 + 9^2) = √82

cos(θ) is the ratio between the adjacent side and the hypotenuse: 1/√82

Using a calculator, you can verify that cos(arctan(9)) = 1/√82