give the exact value of the expression withou...
give the exact value of the expression without using a calculator. cos (tan^(-1)(-3)) cos (tan^(-1)(-3)) = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expressi)
Answer
# Explanation:
## Step1: Let $\theta=\tan^{-1}(-3)$
This means $\tan\theta=-3=\frac{-3}{1}$, and $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$.
## Step2: Consider a right - triangle
If $\tan\theta=\frac{y}{x}=-3$, we can let $y = - 3$ and $x = 1$. Then, by the Pythagorean theorem $r=\sqrt{x^{2}+y^{2}}=\sqrt{1^{2}+(-3)^{2}}=\sqrt{1 + 9}=\sqrt{10}$.
## Step3: Find $\cos\theta$
Since $\cos\theta=\frac{x}{r}$, substituting $x = 1$ and $r=\sqrt{10}$, we get $\cos\theta=\frac{1}{\sqrt{10}}=\frac{\sqrt{10}}{10}$.
# Answer:
$\frac{\sqrt{10}}{10}$