(Hint:Put$\;tan x=t^2$)
$\begin{array}{1 1} (A) \large\frac{1}{\sqrt 2}\normalsize \tan^{-1}\bigg(\large\frac{\tan x+\cot x}{\sqrt 2}\bigg)+\frac{1}{2\sqrt 2}\normalsize\log\begin{vmatrix} \large\frac{\tan x+\cot x-\sqrt 2}{\tan x+\cot x+\sqrt 2}\end{vmatrix}+c \\ (B) \large\frac{1}{\sqrt 2}\normalsize \tan^{-1}\bigg(\large\frac{\tan x-\cot x}{\sqrt 2}\bigg)+\frac{1}{2\sqrt 2}\normalsize\log\begin{vmatrix} \large\frac{\tan x+\cot x-\sqrt 2}{\tan x+\cot x-\sqrt 2}\end{vmatrix}+c \\ (C) \large\frac{1}{\sqrt 2}\normalsize \tan^{-1}\bigg(\large\frac{\tan x-\cot x}{\sqrt 2}\bigg)+\frac{1}{2\sqrt 2}\normalsize\log\begin{vmatrix} \large\frac{\tan x-\cot x-\sqrt 2}{\tan x-\cot x+\sqrt 2}\end{vmatrix}+c \\ (D) \large\frac{1}{\sqrt 2}\normalsize \tan^{-1}\bigg(\large\frac{\tan x-\cot x}{\sqrt 2}\bigg)+\frac{1}{2\sqrt 2}\normalsize\log\begin{vmatrix} \large\frac{\tan x+\cot x-\sqrt 2}{\tan x+\cot x+\sqrt 2}\end{vmatrix}+c \end{array} $
