Question

Prove? `secx/sinx-cscx/secx=tanx`

Answer 1

See below:

Explanation:

`secx/sinx-cscx/secx=tanx`

remember that `secx=1/cosx and cscx=1/sinx`

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

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

`1/(sinxcosc)-cos^2x/(sinxcosx)=tanx`

`(1-cos^2)/(sinxcosc)=tanx`

remember that `sin^2x+cos^2x=1 => sin^2x=1-cos^2x`

`sin^2x/(sinxcosc)=tanx`

`sinx/cosx=tanx`

`tanx=tanx color(white)(000)color(green)root`

Answer 2

`LHS=sec(x)/sin(x) - csc(x)/sec(x)`

`=(sec(x)sec(x) - csc(x)sin(x))/(sinxsecx)`

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

`=tan^2x/tanx`

`= tan(x)=RHS`