How can this be reduced to the simplest form?

(1+tan^2x) / (1+cot^2x)

2 Answers
Mar 6, 2018

See explanation

Explanation:

We want to simplify

(1+tan^2(x))/(1+cot^2(x))

Use the pythagorean trig identity

sin^2(x)+cos^2(x)=1

=>1+cot^2(x)=csc^2(x)color(green)(larr "divided by" sin^2(x))

=>1+tan^2(x)=sec^2(x)color(green)(larr "divided by" cos^2(x))

Thus

(1+tan^2(x))/(1+cot^2(x))=sec^2(x)/csc^2(x)=sin^2(x)/cos^2(x)=tan^2(x)

Mar 6, 2018

tan^2x

Explanation:

"using the "color(blue)"trigonometric identities"

•color(white)(x)1+tan^2x=sec^2x

•color(white)(x)1+cot^2x=csc^2x

•color(white)(x)secx=1/cosx" and "cscx=1/sinx

rArr(1+tan^2x)/(1+cot^2x)

=sec^2x/csc^2x

=(1/cos^2x)/(1/sin^2x)

=1/cos^2x xxsin^2x

=sin^2x/cos^2x

=tan^2x