Question

prove that 1+sec theta - tan theta /1+sec theta +tan theta =1-sin theta / cos theta

Answer 1

Since We Know That Sec²x - Tan²x = 1Then (Secx - Tanx)(Secx + Tanx) = 1{ From (a²-b²) = (a+b)(a-b) }Then Secx - Tanx = 1/(Secx + Tanx)→The Eq. (1 + Secx - Tanx)/(1 + Secx + Tanx) Can be written as (1 + 1/(Secx + Tanx))/(1 + Secx + Tanx)Now Solving Numerator : 1 + 1/(Secx + Tanx) = (Secx + Tanx + 1)/(Secx + Tanx)Now Shifting (Secx + Tanx) To Denominator : (Secx + Tanx + 1)/((Secx + Tanx)× (Secx + Tanx + 1))Secx + Tanx + 1 gets Cancelled.....!!!!!→1/(Secx + Tanx) = Secx - Tanx. (Refer First Three Steps if you didn`t get this :) )→1/Cosx - Sinx/Cosx Since Denominators Are same (Cosx) We can Add the Numerators→(1 - Sinx)/Cosx.