Question

How do you simplify $ \dfrac{{\left( {1 + \tan x} \right)}}{{\left( {1 - \tan x} \right)}} $ ?

Answer 1

Hint: For solving this question, we just need knowledge of some basic trigonometric formulas and relations. We are going to use two simple formulas.
 $
   \Rightarrow \tan \left( {\dfrac{\pi }{4}} \right) = 1 \\
   \Rightarrow \tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}} \;
  $
First of all, we will be substituting 1 with $ \tan \left( {\dfrac{\pi }{4}} \right) $ and then adjust the terms in such a way that we can compare the given equation with the above formula.

Complete step by step solution:
In this question, we are supposed to simplify
  $ \dfrac{{\left( {1 + \tan x} \right)}}{{\left( {1 - \tan x} \right)}} - - - - - - - \left( 1 \right) $
To solve this question, we are going to make some modifications in the above equation.
First of all, we know that the value of $ \tan \left( {\dfrac{\pi }{4}} \right) = $ 1.
So, substitute 1 in equation (1) with $ \tan \left( {\dfrac{\pi }{4}} \right) $
Equation (1) will become
 $ \dfrac{{\left( {1 + \tan x} \right)}}{{\left( {1 - \tan x} \right)}} = \dfrac{{\tan \left( {\dfrac{\pi }{4}} \right) + \tan x}}{{\tan \left( {\dfrac{\pi }{4}} \right) - \tan x}} $
Here, the coefficient of $ \tan x $ is 1.
So, we can write (1) $ \tan x $ instead of $ \tan x $ in the denominator.
So, equation (1) will become,
 $ \dfrac{{\left( {1 + \tan x} \right)}}{{\left( {1 - \tan x} \right)}} = \dfrac{{\tan \left( {\dfrac{\pi }{4}} \right) + \tan x}}{{\tan \left( {\dfrac{\pi }{4}} \right) - \left( 1 \right)\tan x}} - - - - - - - \left( 2 \right) $
Now, we need to substitute 1 with $ \tan \left( {\dfrac{\pi }{4}} \right) $ in equation (2) too.
So, equation (2) becomes
 $ \dfrac{{\left( {1 + \tan x} \right)}}{{\left( {1 - \tan x} \right)}} = \dfrac{{\tan \left( {\dfrac{\pi }{4}} \right) + \tan x}}{{\tan \left( {\dfrac{\pi }{4}} \right) - \tan \left( {\dfrac{\pi }{4}} \right)\tan x}} $
Now, we can write $ \tan \left( {\dfrac{\pi }{4}} \right) $ as 1 in the denominator term.
 $ \dfrac{{\left( {1 + \tan x} \right)}}{{\left( {1 - \tan x} \right)}} = \dfrac{{\tan \left( {\dfrac{\pi }{4}} \right) + \tan x}}{{1 - \tan \left( {\dfrac{\pi }{4}} \right)\tan x}} - - - - - - - \left( 3 \right) $
So, all the modifications are done now and now we can compare the two equations.
Now, we know the formula
 $ \tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}} $
Comparing equation (3) with this formula we get,
 $
  A = \tan \dfrac{\pi }{4} \\
  B = \tan x \;
  $
So, therefore equation (3) becomes
 $ \dfrac{{\left( {1 + \tan x} \right)}}{{\left( {1 - \tan x} \right)}} = \tan \left( {x + \dfrac{\pi }{4}} \right) $
Hence, our final answer is $ \dfrac{{\left( {1 + \tan x} \right)}}{{\left( {1 - \tan x} \right)}} = \tan \left( {x + \dfrac{\pi }{4}} \right) $ .
So, the correct answer is “ $ \tan \left( {x + \dfrac{\pi }{4}} \right) $ .”.

Note: Trigonometric questions are always formula based. So, remember that you need to learn each and every trigonometric formula and always keep them in mind while solving trigonometric questions. Sometimes you have to use relations between them too to substitute the values in the question and get the solution.