Prove the identity: (cos x / (1 - sin x)) - sec x = tan x

We need to prove that :

cosx/
(1-sinc) - sec x = tanx

We will start from the left side
and prove the right side.

First, we know that sec x =
1/cosx

==> cosx / (1-sinx) -
1/cosx

==> Now we will rewrite using a common
denominator cosx(1-sinx)

==> ( cosx*cosx - (1-sinx)
/ cosx(1-sinx)

==> (cos^2 x + sinx -1) /
cosx(1-sinx)

==> We know that cos^2 x = 1- sin^2
x

==> (1-sin^2 x + sinx -1 ) /
cosx(1-sinx)

==> Now we will
factor:

==> (1-sinx)(1+sinx) - (1-sinx) /
cosx(1-sinx)

We will factor
(1-sinx)

==> (1-sinx)[ (1+sinx -1) /
cosx(1-sinx)

Now we will reduce
1-sinx

==> sinx/ cosx =
tanx...........q.e.d

Then we proved that
cosx/ (1-sinx) - sec x = tanx.