cscx-cosxcotx=sinx show it is an identity

Question

the question asks to prove using steps to make the left side look like the right hand side

Answer 1

We know that sin^2x+cos^2x =1

cotx = cosx/sinx

Left hand side

= cscx-cscxcotx

= (1/sinx)-cosx*cotx

= (1/sinx)-cosx*cotx

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

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

= [1-cos^2x*(sinx/sinx)]/sinx

= (1-cos^2x)/sinx

= sin^2x/sinx

= sinx

= Right hand side.