Question

Simplify?

`secx/sinx`-`sinx/cosx`

Answer 1

Write everything in terms of `sinx` and `cosx`. These are the identities you need to use:

`secx=1/cosx`

`cotx=cosx/sinx`

`cos^2x+sinx^2=1`

`=>cos^2x=1-sin^2x`

Here's the actual problem:

`color(white)=secx/sinx-sinx/cosx`

`=(1/cosx)/sinx-sinx/cosx`

`=((1/cosx)*cosx)/(sinx*cosx)-(sinx*sinx)/(sinx*cosx)`

`=((1/color(red)cancelcolor(black)cosx)*color(red)cancelcolor(black)cosx)/(sinx*cosx)-(sinx*sinx)/(sinx*cosx)`

`=1/(sinx*cosx)-(sin^2x)/(sinx*cosx)`

`=(1-sin^2x)/(sinx*cosx)`

`=cos^2x/(sinx*cosx)`

`=cos^color(red)cancelcolor(black)2x/(sinx*color(red)cancelcolor(black)cosx)`

`=cosx/sinx`

`=cotx`

Answer 2

`secx/sinx-sinx/cosx=`

`1/(cosxsinx)-sinx/cosx=`

`1/(cosxsinx)-(sinx(sinx))/(cosx(sinx))=`

`(1-sin^2x)/(cosxsinx)=`

`cos^2x/(cosxsinx)=`

`cosx/sinx=`

`cotx`

Answer 3

The answer is `cot(x)`.

Explanation:

`sec(x)/sin(x)-sin(x)/cos(x)`

Using `sec(t)=1/cos(t)`, transform the expression.
`=1/cos(x)/sin(x)-sin(x)/cos(x)`

Simplify the complex fraction.
`=1/cos(x)sin(x)-sin(x)/cos(x)`

Write all numerators above the least common denominator `cos(x)sin(x)`.
`=1-sin(x)^2/cos(x)sin(x)`

Using `1-sin(t)^2=cos(t)^2`, simplify the expression.
`=cos(x)^2/cos(x)sin(x)`

Reduce the fraction with `cos(x)`
`=cos(x)/sin(x)`

Using `cos(t)/sin(t)=cot(t)`, transform the expression.
`=cot(x)`

I know this is freaking messy, sorry :)