How to verify this identity?(sin 2x / sinx) thank you!

sin2x/sinx - cos2x/cosx =
secx

We will start from the L.H.S and prove the
identity.

We know that:

 sin2x
= 2sinx*cosx

cos2x = 2cos^2 x -
1

We will substitute in
L.H.S.

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

==> 2cosx - 2cos^2 x/cosx  +
1/cosx

==> 2cosx - 2cosx +
1/cosx

Reduce 2cosx

Now we
know that secx = 1/cosx

==> 1/cosx = sec
x.........R.H.S

Then the identity "(sin 2x /
sinx) - (cos 2x / cos x) = sec x" is TRUE.