(i) Show that the equation 3 sin x tan x = 8 can be written as 3 cos^2 x + 8 cos x − 3 = 0.

(ii) Hence solve the equation 3 sin x tan x = 8 for 0◦ ≤ x ≤ 360◦ .

3 sin x tan x = 8
3 sin x (sin x/cos x) = 8
3(sin^2 x/cos x) = 8
3 sin^2 x = 8 cos x
3 (1-cos^2 x) = 8 cos x
3 - 3 cos^2 x = 8 cos x
0 = 3 cos^2 x + 8 cos x -3
Now factor and solve for cos x
0 = (3 cos x-1) (cos x + 3)
cos x = 1/3 or -3
Since cos x can only be between -1 and 1, disregard the -3.
cos x = 1/3
arccos(cos x) = arccos(1/3)
x = 70.5

Test it.
3 sin(70.5) tan(70.5)
3*.9426*2.8239
7.99
Close enough. You can say its equal to 8 due to minor variance in floating point math.

substitute tan x = sin x / cos x

3 sin x sin x / cos x =8
3 sin[x]^2 = 8 cos x

(sin x)^2 = 1 - (cos x)^2

3 (1- cos^2 x) = 8 cos x

3 -3 cos^2 x = 8 cos x
3 cos^2 x+8 cos x -3 = 0