What is cos x, if x is in interval (pi/2 ,pi) and tan x=-12/5?

We'll apply Pythagorean
identity:

(tan x)^2 + 1 = 1/(cos
x)^2

(cos x)^2 = 1/[(tan x)^2 +
1]

We'll plug in the given value of tan
x:

(cos x)^2 = 1/[(-12/5)^2 +
1]

(cos x)^2 = 1/(144/25 +
1)

(cos x)^2 =
25/(144+25)

(cos x)^2 =
25/169

cos x = +sqrt (25/169) or cos x = -sqrt
(25/169)

Since the values of cosine function are negative
in the second quadrant (`pi` /2 ; `pi` ), we'll keep only the negative value for cos
x.

cos x = -
5/13

The requested value for cos x is: cos x
= - 5/13.