Question

Prove that tanx/1+sec x - tan x/1-sec x=2 cosec x

Answer 1

I will assume you meant:

tanx/(1+sec x) - tan x/(1-sec x) = 2 cosec x

the way you typed it would be a false statement

LS = [ tanx(1 - secx) - tanx(1 + secs) ] / ( (1+secx)(1 - secx) )
= [tanx - tanxsecs - tanx - tanxsecx]/(1 - sec^2 x)
= -2tanxsecx / -tan^2 x
= 2 secx/tanx
= 2(1/cosx) / (sinx/cosx)
= 2/sinx
= 2 cscx
= RS