Verify `(secx-tanx)(cscx+1)=cotx` :

Rewrite all terms in terms of sin and cos:

`=(1/(cosx)-(sinx)/(cosx))(1/(sinx)+1)` combine fractions:

`=((1-sinx)/(cosx))((1+sinx)/(sinx))` multiply fractions:

`=(1-sin^2x)/(cosxsinx)` use the pythagorean identity:

`=(cos^2x)/(cosxsinx)` cancel common factor:

`=cosx/sinx`

`=cotx` as required.