find the exact value of each expression: tan(cos^(-1)(-1/sqrt(5))) =

Explanation:

Step1: Let $\theta=\cos^{-1}\left(-\frac{1}{\sqrt{5}}\right)$

This means $\cos\theta =-\frac{1}{\sqrt{5}}$, and $\theta\in[0,\pi]$.

Step2: Use the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$

We get $\sin\theta=\sqrt{1 - \cos^{2}\theta}=\sqrt{1-\left(-\frac{1}{\sqrt{5}}\right)^{2}}=\sqrt{1-\frac{1}{5}}=\sqrt{\frac{4}{5}}=\frac{2}{\sqrt{5}}$ (since $\sin\theta\geq0$ for $\theta\in[0,\pi]$).

Step3: Recall the definition of tangent

$\tan\theta=\frac{\sin\theta}{\cos\theta}$. Substitute $\sin\theta=\frac{2}{\sqrt{5}}$ and $\cos\theta =-\frac{1}{\sqrt{5}}$ into the formula. Then $\tan\theta=\frac{\frac{2}{\sqrt{5}}}{-\frac{1}{\sqrt{5}}}=- 2$.

Answer:

$-2$