cotx/secx-tanx

To simplify the expression cot(x)/sec(x) - tan(x), we need to first understand the properties of trigonometric functions.

We know that:

1. cot(x) = cos(x)/sin(x)
2. sec(x) = 1/cos(x)
3. tan(x) = sin(x)/cos(x)

Now let's substitute these values into the expression:

cot(x)/sec(x) - tan(x)

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

To combine the terms, we need a common denominator. In this case, the denominators are sin(x) and cos(x).

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

So, let's simplify each term individually:

cos(x)/sin(x) = cos(x)*cos(x) / (sin(x)*cos(x)) = cos^2(x) / sin(x)cos(x)

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

sin(x)/cos(x) = sin(x) / (cos(x)/sin(x)) = sin^2(x) / sin(x)cos(x)

Now we can substitute these simplified terms back into the expression:

(cos^2(x) / sin(x)cos(x)) / (sin(x)cos(x) / sin(x)cos(x)) - (sin^2(x) / sin(x)cos(x))

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

Using the difference of squares identity, cos^2(x) - sin^2(x) = cos(2x), we can simplify further:

cos(2x) / sin(x)cos(x)

Since cos(2x) = 2cos^2(x) - 1, we can continue simplifying:

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

Now we have the simplified expression for cot(x)/sec(x) - tan(x):

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

Remember to always double-check your simplifications and identities as you go along.