Find the exact value of costan−1−2.

The given expression is cos(tan^-1(-2)). Here are the steps to solve it:

Step 1: Let's denote θ = tan^-1(-2). This means that tan(θ) = -2.

Step 2: Draw a right triangle that represents this relationship. You can consider the opposite side as -2 (since tan(θ) is negative) and the adjacent side as 1 (you can choose any value, but 1 is the simplest).

Step 3: Using the Pythagorean theorem, you can find the hypotenuse of this triangle. The theorem states that (opposite side)^2 + (adjacent side)^2 = (hypotenuse)^2. So, (-2)^2 + 1^2 = hypotenuse^2, which simplifies to 4 + 1 = hypotenuse^2, so the hypotenuse is sqrt(5).

Step 4: The cosine of an angle in a right triangle is defined as (adjacent side)/(hypotenuse). So, cos(θ) = 1/sqrt(5).

So, the exact value of cos(tan^-1(-2)) is 1/sqrt(5).