Can (sec x - cosec x) / (tan x - cot x) be simplified further?

Given the expression ( sec x - csec x ) / (tan x - cot
x)

We need to simplify.

We
will use trigonometric identities to simplify.

We know
that:

sec x = 1/cos x

csec x =
1/sin x

tan x = sinx/cosx

cot
x = cos x/ sin x

We will substitute into the
expression.

==> (1/cos x - 1/sin x ) / (sin x / cos
x - cos x/ sin x)

==>[ ( sin x - cos x) / sinx*cosx]
/ [ (sin^2 x - cos^2 x)/ cosx*sinx

We will reduce
similar:

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

Now we will simplify the
denominator.

==> (sin x - cos x) / (sin x - cos
x)(sinx + cos x)

Reduce similar
terms.

==> 1/ (sin x + cos
x)

Then the expression ( sec x - csec
x)/(tanx - cot x) can be written as : 1/(sin x + cos
x)