prove identity, cotx/1+cscx=cscx-1/cotx

Question

I need to prove this identity, with full steps.

 

Answer 1

cotx /1+cscx =cscx-1/cotx

We know that csc^2x-cot^2x =1

Left hand side

= cotx /1+cscx

multiple by 1-cscx/1-cscx

= (cotx /1+cscx)(1-cscx/1-cscx)

= cotx(1-cscx)/1^2-csc^2x

= cotx(1-cscx)/1-csc^2x

= cotx(1-cscx)/-cot^2x

= 1-cscx/-cotx

Multiple by negitive one.

= cscx-1/cotx

= Right hand side