trignometric identities: prove 1/sinθ-sinθ≡cosθ/tanθ

We need to prove that
:

1/sin(x) - sinx/tanx = cosx/
tanx

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

==> cos x/
tanx

But we know that tan(x)= sinx /
cosx

We willl
substitute:

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

But we know that cos^2 x = 1- sin^2
x

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

==> Then we prove that
cosx/ tanx = 1/sinx - sinx