Question

How to verify ((csc^(3)x-cscxcot^(2)x))/(cscx) =1 ?

Answer 1

The strategy I used is to write everything in terms of `sin` and `cos` using these identities:

`color(white)=>cscx=1/sinx`

`color(white)=>cotx=cosx/sinx`

I also used a modified version of the Pythagorean identity:

`color(white)=>cos^2x+sin^2x=1`

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

Now here's the actual problem:

`(csc^3x-cscxcot^2x)/(cscx)`

`((cscx)^3-cscx(cotx)^2)/(1/sinx)`

`((1/sinx)^3-1/sinx*(cosx/sinx)^2)/(1/sinx)`

`(1/sin^3x-1/sinx*cos^2x/sin^2x)/(1/sinx)`

`(1/sin^3x-cos^2x/sin^3x)/(1/sinx)`

`((1-cos^2x)/sin^3x)/(1/sinx)`

`(sin^2x/sin^3x)/(1/sinx)`

`(1/sinx)/(1/sinx)`

`1/sinx*sinx/1`

`1`

Hope this helps!

Answer 2

Please see below.

Explanation:

`LHS=(csc^3x-cscx*cot^2x)/cscx`

`=csc^3x/cscx-(cscx*cot^2x)/cscx`

`=csc^2x-cot^2x`

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

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

`=sin^2x/sin^2x=1=RHS`