How can this be reduced to the simplest form?

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

2 Answers
Øko
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)#

Jim G.
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#

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