Q : (sin x + 1 )/ cos x = (((tan x + sec x) -1) / ((tan x + 1) - sec x)) * (((tan x + sec x) + 1) / ((tan x + 1) + sec x))
Let we take RHS to prove LHS,
(((tan x + sec x) -1) / ((tan x + 1) - sec x)) * (((tan x + sec x) + 1) / ((tan x + 1) + sec x)) = ((tan x + sec x)^2 -1) / ((tan x + 1)^2 - sec^2 x)
=(tan^2 x + 2tan x sec x + sec ^2 x -1) / (tan^2 x + 2tan x + 1 - sec^2 x
rearrange,
=(tan^2 x + 2tan x sec x +( sec ^2 x -1)) / (tan^2 x + 1 + 2tan x - (sec^2 x)
replace sec ^2 x -1 = tan^2 x, sec ^2 x = tan^2 x + 1 in the equation
=(tan^2 x + 2tan x sec x + tan^2 x) / (tan^2 x + 1 + 2tan x - (tan^2 x + 1)
=(2tan^2 x + 2tan x sec x ) / ( 2tan x )
= tan x + sec x = sin x/ cos x + 1/cos x
= (sin x + 1)/cos x -> proved