prove that cotx/(cosecx+1)+(cosecx+1)/cotx=2 secx

1+cot^2(x)

cot^2(x)

So
cot(x)(csc(x)-1)/(csc^2(x)-1) = cot(x)(csc(x)-1)/(cot^2(x)) =
(csc(x)-1)/cot(x)

so our left hand side
becomes

Now we can
add because they have the same denominator.  We
get:

simplifying the
top
To simplify I use the inverse function definitions
and to
get

2*1/sin(x)*sin(x)/cos(x) = 2/cos(x) = 2sec(x)

Which is
what we wanted to prove

(csc(x)+1)/cot(x) = 2sec(x)