simplify: (sinx/cosx)-1/(cosx/sinx)-1

I too am slightly frustrated, since I can't tell which order of operation to follow with your placement of brackets

is it sinx/cosx - [1/(sinx/cosx)] - 1 ??

if so, you have

tanx - 1/cotx - 1
= tanx - tanx - 1
= -1

(sinx/cosx-1)/ (cosx/sinx-1)

(tan x -1) / (1/tan x - 1)

(tan^2 x - tan x) / (1 - tan x)

tan x (tan x - 1)/(1 - tan x)

-tan x

To simplify the expression (sin(x) / cos(x)) - 1 / (cos(x) / sin(x)) - 1, we can follow these steps:

Step 1: Simplify the numerator of the first fraction.
The numerator of the first fraction is sin(x).

Step 2: Simplify the denominator of the first fraction.
The denominator of the first fraction is cos(x).

Therefore, the first fraction simplifies to sin(x) / cos(x).

Step 3: Simplify the numerator of the second fraction.
The numerator of the second fraction is 1.

Step 4: Simplify the denominator of the second fraction.
The denominator of the second fraction is cos(x) / sin(x). To simplify this, we can multiply the fraction by the reciprocal (or flip the fraction).

Reciprocal of cos(x) / sin(x) is sin(x) / cos(x).

So, the second fraction becomes 1 / (sin(x) / cos(x)).

Step 5: Combine the fractions.
Now that we have both fractions simplified, we can combine them.

(sin(x) / cos(x)) - 1 / (sin(x) / cos(x)) can be written as a single fraction with a common denominator.

To find the common denominator, we need to multiply the denominators together.
The denominator of the first fraction is cos(x) and the denominator of the second fraction is sin(x) / cos(x).

The common denominator is cos(x) * (sin(x) / cos(x)) = sin(x).

Now, we can write the expression as a single fraction:

(sin(x) - 1) / sin(x).

Therefore, the simplified expression is (sin(x) - 1) / sin(x).