Given: 12sin x 5cos x = 13

To Find: tan x

Explanation:
We can use the trigonometric identity: sin^2 x + cos^2 x = 1

Squaring both sides of the given equation, we get:

(12sin x)^2 + (5cos x)^2 + 2(12sin x)(5cos x) = 169

144sin^2 x + 25cos^2 x + 120sin x cos x = 169

Using the identity, 2sin x cos x = sin 2x, we get:

144sin^2 x + 25cos^2 x + 60sin 2x = 169

Dividing by 25, we get:

(144/25)sin^2 x + cos^2 x + (12/5)sin 2x = 169/25

Using the identity, 1 + tan^2 x = sec^2 x, we get:

(144/25)tan^2 x + 1 + (12/5)tan x = 169/25

Simplifying, we get:

144tan^2 x + 60tan x - 25 = 0

Solving this quadratic equation using the formula, we get:

tan x = (-b ± √(b^2 - 4ac))/2a

tan x = (-60 ± √(60^2 + 4(144)(25)))/2(144)

tan x = (-60 ± √8100)/288

tan x = (-60 ± 90)/288

tan x = -0.576 or 0.139

Answer: The value of tan x is either -0.576 or 0.139.