1) 1-tanx/1-cotx = -tanx

Using the Left side

1-tanx/1-cotx

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

= (cosx-sinx/cosx)/(sinx-cosx/sinx)

= (cosx-sinx)sinx/-(cosx-sinx)cosx

= sinx/-cosx

= - tanx

2) 1-cotx/1-tanx = cotx

Using the Left hand side

1+cotx/1+tanx

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

= (sinx+cosx/sinx)/(cosx+sinx/cosx)

= (sinx+cosx)cosx/(sinx+cosx)sinx

= cosx/sinx

= cotx

Using the equation

tanx=sinx/cosx