91Ó°ÊÓ

Since the angle \(\tan ^{-1}(-4)\) is given, let's call it \(A\). The tangent of angle \(A\) is given by \(-\frac{4}{1}\) or simply \(-4\), meaning the opposite side is -4 units and the adjacent side is 1 unit. Since the lengths of the sides of a triangle cannot be negative, we will use the absolute value of the opposite side (which is 4) to keep our calculations meaningful. The signs are important in the context of trigonometric ratios, so the cosine sign will depend on the quadrant the angle is located. We can create a right triangle with the sides like this: - Opposite side = 4 units - Adjacent side = 1 unit

Now that we have our right triangle, we will use the Pythagorean theorem to find the length of the hypotenuse. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. That is: \[h^2 = a^2 + b^2\] where \(a\) and \(b\) are the legs of the triangle and \(h\) is the hypotenuse. Plugging in the values for the legs, we have: \[h^2 = 4^2 + 1^2\] \[h^2 = 16 + 1\] \[h^2 = 17\] Taking the square root of the result gives us the length of the hypotenuse: \[h = \sqrt{17}\]

We can now use the definition of cosine, which is \(\frac{adjacent}{hypotenuse}\), to evaluate \(\cos \left( \tan ^{-1}(-4) \right)\). So we have: \[\cos \left( \tan ^{-1}(-4) \right) = \frac{1}{\sqrt{17}}\] Let's not forget about the sign of the cosine. In this case, since we took the absolute value of the opposite side, we know that angle \(A\) is in the second or fourth quadrant as the tangent is negative in these quadrants. The cosine is positive in the fourth quadrant and negative in the second quadrant. Since the inverse tangent function has a range of \((-\frac{\pi}{2}, \frac{\pi}{2})\), we know that the angle \(A\) must be in the second quadrant (if expressed in radians) or the fourth quadrant if expressed in degrees. As a result, we can conclude that the cosine of the angle is negative. So, the final answer is: \[\cos \left( \tan ^{-1}(-4) \right) = - \frac{1}{\sqrt{17}}\]