Question

Prove the following ( sin x - cos x +1)/ (sin x + cos x -1) = 1 / (sec x - tan x )
Please do explain it ..reply as soon as possible....

Answer 1

Answer:

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

Step-by-step explanation:

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

LHS =

( sin x - cos x +1)/ (sin x + cos x -1)

Multiplying & Dividing by (sin x + cos x + 1)

= {(Sin x+ 1)² - Cos²x } / { (sin x + cos x)² - 1² }

= (Sin²x + 1 + 2Sinx - Cos²x) / (sin²x + cos²x  + 2SinxCosx - 1)

using sin²x + cos²x = 1 or 1 - cos²x = Sin²x

= (Sin²x + 2Sinx + Sin²x) / (1  + 2SinxCosx - 1)

= (2Sin²x  + 2Sinx) /(2SinxCosx)

= (Sinx  + 1) /Cosx

Multiplying & Dividing by 1 - Sinx

= (1 - Sin²x) /(Cosx (1 - Sinx))

= Cos²x/(Cosx (1 - Sinx))

= Cosx/(1 - Sinx)

= 1/(1/Cosx - Sinx/Cosx)

= 1/(Secx - tanx)

= RHS

QED

Proved

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