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.