Find \sin(2x), \cos(2x), and \tan(2x) from the given information.
\csc(x) = 5, \tan(x) < 0
Step 1
If \csc(x) = 5, then \sin(x) = \boxed{\frac{1}{5}}. Since sine is opposite over hypotenuse, you can find the adjacent leg, a, using the Pythagorean Theorem.
$a^2 + 1^2 = \boxed{5^2}$
$a^2 = \boxed{24}$
a = \boxed{2\sqrt{6}}$
So, you have \cos(x) = \boxed{-\frac{2\sqrt{6}}{5}} because \tan(x) < 0.