Question

tanx cotx
------- + ------- = 1 +secxcscx
1 - cotx 1- tanx

prove the proof

Answer 1

tanx/(1-cotx) + cotx/(1-tanx)
= tanx/(1-cotx) + cotx/(1-tanx) * (-tanx/-tanx)
= tanx/(1-cotx) - tan^2x/(1-tanx)
= (cotx-tan^2x)/(1-tanx)
= (1-tan^3x)/(tanx(1-tanx))
= (1-tanx)(1+tanx+tan^2x) / (1-tanx)(tanx)
= (1+tanx+tan^2x)/tanx
= 1/tanx + 1 + tanx
= 1 + sinx/cosx + cosx/sinx
= 1 + (sin^2x+cos^2x)/(sinx cosx)
= 1 + secx cscx

Answer 2

oops . typo in line 2, but you should be able to catch it.