simplify trig expression

[detectivea0149]

(1+tanx)/(1+cotx)*(csc^​2x-1)

=cos/cos+(sin/cos)/(sin/sin)+(cos/sin)*(1/sin^(2)x)
=sin/cos(1/sin²x-1)
then I'm not sure what to do next

 

[Deleted member 4993]

(1+tanx)/(1+cotx)*(csc^​2x-1)

=cos/cos+(sin/cos)/(sin/sin)+(cos/sin)*(1/sin^(2)x)
=sin/cos(1/sin²x-1)
then I'm not sure what to do next

I assume the expression to be simplified is:

\(\displaystyle \displaystyle{\frac{1+tan(x)}{1+cot(x)} * (csc^2(x)-1)}\)

quickest way to solve this is to recall:

cot(x) = 1/tan(x) and

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

then:

\(\displaystyle \displaystyle{\frac{1+tan(x)}{1+cot(x)} * (csc^2(x)-1)}\)

\(\displaystyle = \ \displaystyle{\frac{1+tan(x)}{1+\frac{1}{tan(x)}} * cot^2(x)}\)

\(\displaystyle = \ \displaystyle{tan(x) * cot^2(x)}\)

= cot(x)