How do you simplify (1+tan^2x)/(1+cot^2x)?

##tan^2x##
Start with simplifying ##tan^2x## and ##cot^2x## :
##(1+sin^2x/cos^2x)/(1+cos^2x/sin^2x)##
this is the same as:
##(1+sin^2x/cos^2x) * (1/(1+cos^2x/sin^2x))##
##1## in the first parantheses can be rewritten as:
##cos^2x/cos^2x##
Similarly the ##1## in the denominator of the second parentheses can be rewritten as:
##sin^2x/sin^2x##
leaving you with
##((cos^2x+sin^2x)/cos^2x)* (1/((sin^2x+cos^2x)/sin^2x))##
Using the Pythagorean Identity we are left with:
##(1/cos^2x) * (1/(1/sin^2x))##
Simplifies to:
##1/cos^2x * sin^2x## = ##sin^2x/cos^2x##
Using the definition of tangent:
##sin^2x/cos^2x## = ##tan^2x##