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##