Solve the given integral $ \int {\dfrac{{{{\sec }^2}\theta }}{{\tan \theta }}d\theta } $

Hint: This question is of integration. Integration has many formulas and many methods to solve the problem. Given a problem contains trigonometric ratios we can go about solving the problem by reducing the problem into simple form.

Complete step-by-step answer:
Given,
$ \int {\dfrac{{{{\sec }^2}\theta }}{{\tan \theta }}d\theta } $
As we know,
$
  \sec \theta = \dfrac{1}{{\cos \theta }} \\
  \tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }} \;
 $
From the given problem
$
   = \dfrac{{{{\sec }^2}\theta }}{{\tan \theta }} = \dfrac{{\cos \theta }}{{\sin \theta \times {{\cos }^2}\theta }} \\
   \Rightarrow \dfrac{1}{{\sin \theta \cos \theta }} \;
    \\
 $
Since,
$ \sin 2\theta = 2\sin \theta \cos \theta $
We get,
$ = \dfrac{{{{\sec }^2}\theta }}{{\tan \theta }} = \cos ec2\theta $
Therefore, we can write our problem as
$
   = \int {\cos ec2\theta \,d\theta } \\
   \Rightarrow \dfrac{{ - \ln \left| {\cos ec2\theta + \cot 2\theta } \right|}}{2} + C \;
 $
The above is the answer to the given question.
So, the correct answer is “$\dfrac{{ - \ln \left| {\cos ec2\theta + \cot 2\theta } \right|}}{2} + C $”.

Note: Students need to know the trigonometric identities and it’s conversions. It allows reduction of a given problem into the simple form and then integrates to get the solution.
Mostly they reduce to the form that has its standard integral solution.