If cos x=1/9,find all possible values of sec x-tan x/sin x,by showing all the working out,

tan x = sin x / cos x

sec x - tan x / sin x =

1 / cos x - ( sin x / cos x ) / sin x =

1 / cos x - sin x / cos x * sin x =

1 / cos x - 1 / cos x = 0

sec x - tan x / sin x is always zero, regardless value of cos x

To find all possible values of the expression sec(x) - tan(x) / sin(x), we can use the given information that cos(x) = 1/9. Let's express the expression in terms of sin(x) and cos(x):

sec(x) - tan(x) / sin(x) = 1/cos(x) - sin(x)/cos(x) / sin(x)

Since cos(x) = 1/9, we substitute it into the expression:

sec(x) - tan(x) / sin(x) = 1/(1/9) - sin(x)/(1/9) / sin(x)

Reciprocal of 1/9 is 9/1, so simplifying further:

sec(x) - tan(x) / sin(x) = 9 - 9sin(x) / sin(x)

Next, we can simplify by factoring out 9 from the numerator:

sec(x) - tan(x) / sin(x) = 9 * (1 - sin(x)) / sin(x)

Since we know that cos(x) = 1/9, we can determine sin(x) using the identity sin^2(x) + cos^2(x) = 1. Substitute cos(x) into the equation:

sin^2(x) + (1/9)^2 = 1
sin^2(x) + 1/81 = 1
sin^2(x) = 1 - 1/81 = 80/81

Taking the square root of both sides:

sin(x) = ± √(80/81)

sin(x) = ± (8/9)√5/9

Now, substitute these values of sin(x) into the expression:

1) When sin(x) = (8/9)√5/9:
sec(x) - tan(x) / sin(x) = 9 * (1 - (8/9)√5/9) / (8/9)√5/9

Simplify the expression:

sec(x) - tan(x) / sin(x) = 9 - 8√5 * 9 / 8√5
sec(x) - tan(x) / sin(x) = 9 - 9
sec(x) - tan(x) / sin(x) = 0

2) When sin(x) = -(8/9)√5/9:
sec(x) - tan(x) / sin(x) = 9 * (1 - (-(8/9)√5/9)) / (-(8/9)√5/9)

Simplify the expression:

sec(x) - tan(x) / sin(x) = 9 * (1 + 8√5 * 9) / (8√5/9)
sec(x) - tan(x) / sin(x) = 9 * (1 + 8√5) / (8√5/9)
sec(x) - tan(x) / sin(x) = 9 * (9 + 8√5) / 8√5

Therefore, the possible values of sec(x) - tan(x) / sin(x) are 0 and 9 * (9 + 8√5) / 8√5.