Find the exact value tan(tan^-1(-1))
The expression tan(tan^(-1)(-1)) represents the tangent of the inverse tangent of -1.
The inverse tangent of -1 is an angle whose tangent is -1. This occurs in the fourth quadrant of the unit circle at an angle of -π/4 radians or -45 degrees.
So, tan(tan^(-1)(-1)) = tan(-π/4).
The tangent function is negative in the second and fourth quadrants, so we can determine the equivalent positive angle by adding π or 180 degrees.
tan(-π/4) = tan(-π/4 + π) = tan(3π/4).
Now, we need to find the exact value of tan(3π/4).
We know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle.
We can construct a right triangle where the angle 3π/4 is formed by the hypotenuse and one of the legs. By the Pythagorean theorem, the lengths of the other two sides can be found.
To construct this triangle, let's assume the hypotenuse has a length of 1. Then, the length of the opposite side is sin(3π/4) and the length of the adjacent side is cos(3π/4).
sin(3π/4) = sqrt(2)/2
cos(3π/4) = -sqrt(2)/2
Therefore, the exact value of tan(3π/4) is sin(3π/4)/cos(3π/4):
tan(3π/4) = (sqrt(2)/2) / (-sqrt(2)/2)
= (sqrt(2)/2) * (-2/sqrt(2))
= -1
So, the exact value of tan(tan^(-1)(-1)) is -1.