Question

Let the trigonometric equations be $\tan A + \tan B + \tan C = 6$ and $\tan A.\tan B = 3$ , then $\tan C$ is equal to
A. 2
B. 1
C. 3
D. -1

Answer 1

Hint: Try to use the supplementary condition of angles of a triangle and the tan sum formula to simplify the given equation. Then use the other given equation to find the desired value.

Complete Step-by-Step solution:
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry.
Here in the question we are given three angles of a triangle $A,B{\text{ }}and{\text{ }}C$.
We also know that the sum of angles of a triangle equals the straight angle or ${180^0}$.
Hence, we have our first condition as
$A + B + C = {180^0}{\text{ -------------- (1)}}$
Also, we know the trigonometric formula as
$\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}{\text{ ----- (2)}}$

In the problem we are given,
$
  \tan A + \tan B + \tan C = 6{\text{ --------------- (3)}} \\
  \tan A.\tan B = 3{\text{ ----------------- (4)}} \\
$

We need to simplify the given equations one by one.
From equation (3)
$\tan A + \tan B + \tan C = 6$
Transposing $\tan C$ on the RHS, we get,
$ \Rightarrow \tan A + \tan B = 6 - \tan C$
Now in order to change the above in the form of equation (2), we divide both sides by $1 - \tan A\tan B$,
$ \Rightarrow \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}} = \dfrac{{6 - \tan C}}{{1 - \tan A\tan B}}$

Now we need to use the above stated trigonometric formula in equation (2) in above
$ \Rightarrow \tan \left( {A + B} \right) = \dfrac{{6 - \tan C}}{{1 - \tan A\tan B}}$
Also, it is given in equation (1) as $A + B + C = {180^0}$,
$ \Rightarrow \left( {A + B} \right) = {180^0} - C$
Using in above, we get
$ \Rightarrow \tan \left( {{{180}^0} - C} \right) = \dfrac{{6 - \tan C}}{{1 - \tan A\tan B}}$
We know that $\tan \left( {{{180}^0} - \theta } \right) = - \tan \theta $, as $\tan $ is negative in the second quadrant.
Also, it is predefined in the problem as stated in equation (4) as $\tan A.\tan B = 3$.
Using these results in the previous equation, we get,
$
   \Rightarrow - \tan C = \dfrac{{6 - \tan C}}{{1 - 3}} \\
   \Rightarrow 2\tan C = 6 - \tan C \\
   \Rightarrow \tan C = \dfrac{6}{3} = 2 \\
$
Hence the value of $\tan C$ is equal to 2.
Therefore, option (A). 2 is the correct answer.

Note: In problems involving product or sum of tan terms like above, the sum or difference angle formula of tan trigonometric ratio is used and is needed to be kept in mind. It is to be noted that the tangent formula is only applicable for angles in the domain of tan trigonometric ratio. The range of tan is the whole number line.