If sec x = -3, 0 < x < pi, find cot x.

sec x = -3

then cos x = -1/3, and x is in the 2nd quadrant
the triangle with angle x is in quad II and has a hypotenuse of 3 and an adjacent of 1, so the opposite is
√(9-1) = √8 or 2√2

cot x = adjacent/opposite = -1/√8
or
-√8/8 = -2√2/8 = -√2/4

To find the value of cot x, we need to recall the definitions of sec x and cot x.

Sec x is the reciprocal of cos x, and cot x is the reciprocal of tan x.

Given that sec x = -3, we know that cos x = -1/3 (reciprocal of -3).

To find cot x, let's first find tan x.

We know that tan x = sin x / cos x.

Now, we need to find sin x.

Using the Pythagorean identity: sin^2 x + cos^2 x = 1, we can solve for sin x.

1 = sin^2 x + (-1/3)^2
1 = sin^2 x + 1/9
8/9 = sin^2 x

Taking the square root of both sides, we get:

sin x = ±√(8/9)

Since 0 < x < π, sin x is positive, so sin x = √(8/9).

Now, we can find tan x:

tan x = sin x / cos x
tan x = (√(8/9)) / (-1/3)
tan x = -√(8/9) * 3
tan x = -3√(8/9)

Finally, to find cot x, we take the reciprocal of tan x:

cot x = 1 / tan x
cot x = 1 / (-3√(8/9))
cot x = -1 / (3√(8/9))

Therefore, cot x = -1 / (3√(8/9)).