Question

We have to solve for the problem below to make sure that it is equivalent to 2csc(x). However, I keep getting 2/2sin(x) and not 2/sin(x), which is what 2csc(x) is.

tan(x)/(1+sec(x))+ (1+sec(x))/tan(x)= 2csc(x)

Answer 1

tan(x)/(1+sec(x))+ (1+sec(x))/tan(x)
= tan x/[(cosx+1)/cosx]
+ [(cosx+1)/cosx]/tanx
= sin x/(cosx+1) + (cosx +1)/sinx
= [sin^2 x + (cosx +1)^2]/[sinx(1 + cosx)]
= (sin^2x + cos^2x + 2 cos x + 1)/[sinx(1 + cosx)]
= 2 (1 + cosx)/[sinx(1 + cosx)]
= 2/sin x = 2 csc x