\[\begin{array}{*{20}{c}}
  {\begin{array}{*{20}{c}}
  {\left[ {\frac{{\operatorname{Sin} 2A}}{{1 + \cos 2A}}} \right] \times }&{\left[ {\frac{{\operatorname{Cos} A}}{{1 + \operatorname{Cos} A}}} \right]}
\end{array}}& =
\end{array}\]
(1) \[\tan \frac{A}{2}\]
(2) \[\cot \frac{A}{2}\]
(3) \[\sec \frac{A}{2}\]
(4) \[\cos ec\frac{A}{2}\]

Hint:This question is from the chapter, named Trigonometry. Apply the formula of Sin2A and cos2A to reduce the trigonometric expression in terms of \[\operatorname{Sin} A\] and \[\operatorname{Cos} A\]. Use all the basic formulas of trigonometry to simplify the trigonometric expression. After that apply the trigonometric ratios that will help to reduce the expression.

Formula Used:
1) \[\begin{array}{*{20}{c}}
  {\operatorname{Sin} 2A}& = &{2\operatorname{Sin} A\operatorname{Cos} A}
\end{array}\]
2) \[\begin{array}{*{20}{c}}
  {\operatorname{Cos} 2A}& = &{2{{\operatorname{Cos} }^2}A - 1}
\end{array}\]

Complete step by step Solution:
There are certain steps involved to solve these kinds of questions. A certain producer should be followed to simplify these kinds of trigonometric expressions.
Our main purpose is to simplify the above expression as much as we can. So, to do that, we will have to use all the basic fundamentals of trigonometry.
 Before moving forward, we will use the formula of sin2A and cos2A to reduce the trigonometric expression in terms of \[\operatorname{Sin} A\]and \[\operatorname{Cos} A\].
Now, we can write.
\[ \Rightarrow \left[ {\frac{{\operatorname{Sin} 2A}}{{1 + \operatorname{Cos} 2A}}} \right]\]\[\left[ {\frac{{\operatorname{Cos} A}}{{1 + \operatorname{Cos} A}}} \right]\] …………………………(A)
We know that,
\[\begin{array}{*{20}{c}}
  { \Rightarrow \operatorname{Sin} 2A}& = &{2\operatorname{Sin} A\operatorname{Cos} A}
\end{array}\] and \[\begin{array}{*{20}{c}}
  {\operatorname{Cos} 2A}& = &{2{{\operatorname{Cos} }^2}A - 1}
\end{array}\]
Therefore, from equation (A). we can write
\[ \Rightarrow \left[ {\frac{{2\operatorname{Sin} A\operatorname{Cos} A}}{{1 + 2{{\operatorname{Cos} }^2}A - 1}}} \right]\]\[\left[ {\frac{{\operatorname{Cos} A}}{{1 + \operatorname{Cos} A}}} \right]\]
\[ \Rightarrow \left[ {\frac{{\operatorname{Sin} A\operatorname{Cos} A}}{{{{\operatorname{Cos} }^2}A}}} \right]\]\[\left[ {\frac{{\operatorname{Cos} A}}{{1 + \operatorname{Cos} A}}} \right]\]
\[ \Rightarrow \left[ {\frac{{\operatorname{Sin} A}}{{1 + \operatorname{Cos} A}}} \right]\] ……………………. (B)
To reduce this trigonometric expression, we will again expand \[\operatorname{Sin} A\]and \[\operatorname{Cos} A\]. For this purpose, we will use the expansion formula of \[\operatorname{Sin} A\]and\[\operatorname{Cos} A\].
So, we know that
\[ \Rightarrow \begin{array}{*{20}{c}}
  {\operatorname{Sin} A}& = &{2\operatorname{Sin} \frac{A}{2}\operatorname{Cos} \frac{A}{2}}
\end{array}\] and \[\begin{array}{*{20}{c}}
  {\operatorname{Cos} A}& = &{2{{\operatorname{Cos} }^2}\frac{A}{2}}
\end{array} - 1\]
Therefore, from equation (B). we can write
\[ \Rightarrow \left[ {\frac{{2\operatorname{Sin} \frac{A}{2}\operatorname{Cos} \frac{A}{2}}}{{1 + 2{{\operatorname{Cos} }^2}\frac{A}{2} - 1}}} \right]\]
\[ \Rightarrow \left[ {\frac{{\operatorname{Sin} \frac{A}{2}}}{{\operatorname{Cos} \frac{A}{2}}}} \right]\]
Now, we know that
\[\begin{array}{*{20}{c}}
  { \Rightarrow \tan \frac{A}{2}}& = &{\frac{{\operatorname{Sin} \frac{A}{2}}}{{\operatorname{Cos} \frac{A}{2}}}}
\end{array}\]
Therefore, we can write
\[ \Rightarrow \tan \frac{A}{2}\]
Now, the final answer is \[\tan \frac{A}{2}\]

Hence, the correct option is 1.

Note: Use all the basic trigonometric formulas to reduce the expression. Also, use the half-angle trigonometric formula and apply these formulas until the expressions get simple. After that use trigonometric ratios. All the formulas that you are going to apply, should be in such a manner that there are no errors in the solution.