Prove that: tanx*sinx / (sec^2 x -1) = cosx

We need to prove
that:

(tanx*sinx) / (sec^2 x -1) =
cosx

We will start from the left side and prove that it
equals cosx.

First we will rewrite using the trigonometric
identities.

We know that secx =
1/cosx

==> sec^2 x = 1/cos^2
x

==> (tanx*sinx)/(sec^2 x -1) =
(tanx*sinx)/(1/cos^2 x  
-1)

                                            =
(tanx*sinx)/[ (1-cos^2 x)/
cos^2x]

                                         = cos^2
x(tanx*sinx) / (1-cos^2 x)

Also, we know that 1-cos^2 x =
sin^2

==> cos^2x(tanx*sinx) / sin^2
x

Now we know that tanx =
sinx/cosx

==> cos^2 x * sinx*sinx / cosx * sin^2
x

==> (cos^2 x * sin^2 x) / cosx * sin^2
x

Now we will reduce
similar.

==> cosx ......
q.e.d