Question

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

Answer 1

L.H.S.= secx+tanx -1/tanx-secx+1

=secx+tanx-(sec^2 x-tan^2 x)/tanx-secx+1 ........ (sec^2 x-tan^2 x=1)

=secx+tanx-(secx+tanx)(secx-tanx)/tanx-secx+1 ......... (a^2-b^2=(a+b)(a-b)

=(secx+tanx)[1-(secx-tanx)]/tanx-secx+1 .......... (taking secx+tanx as common in the numerator)

=(secx+tanx)(1-secx+tanx)/tanx -secx+1

=(secx+tanx)(tanx-secx+1)/(tanx - secx+1)

=secx+tanx

=1/cosx + sinx/cosx

=1+sinx/cosx

Hence proved.