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
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