Use identities to find values of the sine and cosine functions of the function for the angle measure.
\[
2 x, \text { given } \tan x=-3 \text { and } \cos x< 0
\]
\[
\cos 2 x=
\]
(Use fractions or pi for any numbers in the expression.)

Step 1 ：We are given that \(\tan x = -3\) and \(\cos x < 0\). We are asked to find \(\cos 2x\). We know that \(\cos 2x = 1 - 2\sin^2x\) or \(\cos 2x = 2\cos^2x - 1\). However, we are not directly given the values of \(\sin x\) or \(\cos x\). We can find these values using the given information.

Step 2 ：We know that \(\tan x = \frac{\sin x}{\cos x}\). Since \(\tan x = -3\) and \(\cos x < 0\), we know that \(\sin x\) must be positive (because a negative divided by a negative is a positive).

Step 3 ：We can find the value of \(\sin x\) by using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). We know that \(\cos^2 x = \frac{1}{1 + \tan^2 x}\), so we can substitute this into the Pythagorean identity to find \(\sin^2 x\).

Step 4 ：Once we have the value of \(\sin^2 x\), we can substitute it into the identity for \(\cos 2x\) to find the value of \(\cos 2x\).

Step 5 ：Substituting the values, we get \(\tan x = -3\), \(\cos x = -0.31622776601683794\), \(\sin x = 0.9486832980505138\), \(\sin^2 x = 0.8999999999999999\), and \(\cos 2x = -0.7999999999999998\).

Step 6 ：Final Answer: The value of \(\cos 2x\) is approximately \(\boxed{-0.8}\).