Question

How do I simplify (tan x/(1 + sec x))+((1 + sec x)/(tan x))?

Answer 1

the same way you would simplify 2/3 + 3/2. Put it all over a common denominator:

tan/(1+sec) + (1+sec)/tan
= [tan^2 + (1+sec)^2]/(tan(1+sec))
= (tan^2 + 1+2sec+sec^2)/(tan(1+sec))
= (sec^2-1+1+2sec+sec^2)/(tan(1+sec))
= (2sec^2+2sec)/(tan(1+sec))
= 2sec(1+sec)/(tan(1+sec))
= 2sec/tan
= 2/cos * cos/sin
= 2/sin
= 2csc