Verify the identity please read the rest of the question I h

Verify the identity (please read the rest of the question- I have posted this question twice already and gotten the same answers both times. I know that both sides will simplify to cosx, but that is not sufficient for the answer. It must be shown how one side can equal the other by working only on one side.

Here is the problem:

and here is an answer already given:

This answer is not sufficient. Please show how one side can equal the other by only working on one side. Thank you for your help.

Solution

cosx -sinx / 1-tanx = cosx +sinx / 1+tanx

L.H.S = cosx -sinx / 1-tanx

= cosx -sinx / 1 - sinx/cosx

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

= cosx

multiplyand divide by (sinx + cosx)

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

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

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

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

SO L.H.S= R.H.S

hence proved