give the exact value of the expression without using a calculator. \ncos \\left(\\tan ^{-1}(-4)\\right)…

Explanation:

Step1: Let $\theta=\tan^{-1}(-4)$

By the definition of the inverse - tangent function, $\tan\theta=-4$, and $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Since $\tan\theta = - 4<0$, then $\theta\in\left(-\frac{\pi}{2},0\right)$. And we know that $\tan\theta=\frac{y}{x}=-4=\frac{-4}{1}$, so we can consider a right - triangle (in the fourth quadrant) with $y = - 4$ and $x = 1$.

Step2: Find the hypotenuse $r$

Using the Pythagorean theorem $r=\sqrt{x^{2}+y^{2}}$. Substitute $x = 1$ and $y=-4$ into the formula, we get $r=\sqrt{1^{2}+(-4)^{2}}=\sqrt{1 + 16}=\sqrt{17}$.

Step3: Find $\cos\theta$

By the definition of cosine function $\cos\theta=\frac{x}{r}$. Since $x = 1$ and $r=\sqrt{17}$, then $\cos\left(\tan^{-1}(-4)\right)=\cos\theta=\frac{1}{\sqrt{17}}$. Rationalizing the denominator, we multiply the numerator and denominator by $\sqrt{17}$, so $\frac{1\times\sqrt{17}}{\sqrt{17}\times\sqrt{17}}=\frac{\sqrt{17}}{17}$.

Answer:

$\frac{\sqrt{17}}{17}$