We will start from the left side of the equation and try to get right side of equation.

`cscx-(sinx)/(1+cosx)=`  

Now we use definition of cosecant `cscx=1/sinx.`

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

Now we add two fractions together.

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

Now we use basic trigonometric identity `sin^2x+cos^2x=1=>1+sin^2x=cos^2x.`

`(cos^2x+cosx)/(sinx(1+cosx))=`

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

`cosx/sinx=`

Now we use the fact that `cotx=cosx/sinx.`

`cotx.`

We have successfully proved that left side of the equation is equal to the right side of equation.

Q.E.D.