How do you simplify the expression #(1+tan^2t+sec^2tcot^2t)/(csc^2t+cot^2tcsc^2t)#?

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Answer 1 · 2016-12-20T09:46:50

#tan^2t#

To start we need to remember the basic identities.

#1+tan^2x=sec^2x#

#1+cot^2x=csc^2x#

#sin^2x+cos^2x=1#

#(color(red)(1+tan^2t)+sec^2tcot^2t)/(csc^2t+cot^2tcsc^2t)#

#(color(red)(sec^2t)+sec^2tcot^2t)/(csc^2t+cot^2tcsc^2t)#

take out common factors

#(sec^2t[1+cot^2t])/(csc^2t[1+cot^2t]#

cancel where possible

#(sec^2tcancel([1+cot^2t]))/(csc^2tcancel([1+cot^2t])#

now using #" "secx=1/cosx" "&" "cscx=1/sinx#

#sec^2t/csc^2t=(sect/csct)^2#

&#" "tanx=sinx/cosx#

#=((1/cost)/(1/sint))^2#

#=(sint/cost)^2#

#=(tant)^2=tan^2t#