2008/007) Verify (tanx+secx)/(secx-cosx+tanx) = cscx

AS tan x + sec x is in both numerator and dnominator and denominator has one extra term I would take reciprocal of LHS

1/ LHS = 1 - cosx/(tan x + sec x)
= 1- cos x ( sec x- tan x)/ (tan x + sec x)( sec x- tan x)
= 1 - cos x ( sec x - tan x)/ (sec^2 x - tan ^2 x)
= 1- cos x( sec x - tan x) as sec^2x - tan ^2x = 1
= 1 - cos x(1/cosx - sinx /cos x)
= 1-1+ sin x = sin x
so LHS = 1/ sin x = csc x


proved