How do I simplify (tanx/1+secx) + (1+secx/tanx)

Tan x / (1 + sec x) + (1+sec x) / tan x

Tan x=sin x / cos x

1 + sec x=1 + 1/cos x = (cos x+1) / cos x

Therefore:

tan x / (1 + sec x) = (sin x/cos x) / (cos x+1) / cos x=

= (sin x * cos x) / [cos x * (cos x+1) ]=sin x / (Cos x+1)

(1+sec x) / tan x=[ (cos x+1) / cos x] / (sin x/cos x) =

=[cos x (cos x+1) ] / (sin x * cos x) = (cos x+1) / sin x

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

=sin x / (Cos x+1) + (cos x+1) / sin x=

= (sin²x+cos²x+2 cos x+1) / [sin x (cos x+1) ]=

Remember: sin²x+cos²x=1⇒ sin²x=1-cos²x

= (1-cos²x+cos²x+2 cos x+1) / [sin x (cos x+1) ]=

=2 cos x+2 / [sin x (cos x+1) ]=

=2 (cos x+1) / [sin x (cos x+1) ]=

=2 / sin x

Answer : tan x / (1 + sec x) + (1+sec x) / tan x = 2/sin x