Can anyone help me verify this identity?

Question

sin x + cos x= (cos x / 1-tan x) + (sin x / 1-cot x)

Answer 1

Jus do it this way
Multiply the right side by (1-tanx)(1-Cotx).
Then u get Sinx+CosX= Cosx(1-Cotx)+Sinx(1-tanx)/(1-tanx)(1-Cotx).
Now cross multiply
take d denominator on the right side and send it over 2 d left side and u get d same equation on both sides.
Ur equation is verified as d left now equals d right.

Answer 2

RHS
= cos x / (1 - sin x / cos x) + sin x(1 - cos x / sin x)
= cos² x / (cos x - sin x) + sin² x(sin x - cos x)
= cos² x / (cos x - sin x) - sin² x/(cos x - sin x)
= (cos² x - sin² x) / (cos x - sin x)
= (cos x - sin x).(cos x + sin x) / (cos x - sin x)
= (cos x + sin x)
= LHS