tanx+cotx
-----------
sec^2x

Cot x can be written as 1
----
tanx

tanx + 1
----
tan x
---------------
sec^2 x

Simplifying and taking tanx LCM in numerator

(tan^2x + 1)
------------------
tan x
---------------
sec^2 x

or we can rewrite it as

(tan^2x + 1)
------------------
sec^2x * tan x

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

So putting its value in numerator

sec^2x
------------------
sec^2x * tan x

cancelling sec^2x from numerator and denominator

1
--
tanx

which is equal to cotx hence cotx is the answer