Question

If \[{\tan ^{ - 1}}\left( {\dfrac{1}{3}} \right) + {\tan ^{ - 1}}\left( {\dfrac{3}{4}} \right) - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0\], then \[x\] is equal to
A. \[\dfrac{7}{3}\]
B. \[3\]
C. \[\dfrac{{11}}{3}\]
D. \[\dfrac{{13}}{3}\]

Answer 1

Hint: In the given question, we are asked to find the value of \[x\]. For that, we apply the inverse trigonometric function formula and simplify it to get the desired result.

Formula used:
We have been using the following formulas:
1. \[{\tan ^{ - 1}}A + {\tan ^{ - 1}}B = {\tan ^{ - 1}}\left( {\dfrac{{A + B}}{{1 - AB}}} \right)\]

Complete step-by-step solution:
We are given that \[{\tan ^{ - 1}}\left( {\dfrac{1}{3}} \right) + {\tan ^{ - 1}}\left( {\dfrac{3}{4}} \right) - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0\]
We need to find the value of \[x\]
Now we know that \[{\tan ^{ - 1}}A + {\tan ^{ - 1}}B = {\tan ^{ - 1}}\left( {\dfrac{{A + B}}{{1 - AB}}} \right)\]
Now we apply the above inverse trigonometric formulas in the first two terms of the given function, we will get
\[{\tan ^{ - 1}}\left[ {\dfrac{{\dfrac{1}{3} + \dfrac{3}{4}}}{{1 - \dfrac{1}{3} \times \dfrac{3}{4}}}} \right]\, - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0\]
Now we simplify the above equation, we will get
\[
  {\tan ^{ - 1}}\left[ {\dfrac{{\dfrac{{4 + 3 \times 3}}{{12}}}}{{1 - \dfrac{3}{{12}}}}} \right] - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0 \\
  {\tan ^{ - 1}}\left[ {\dfrac{{\dfrac{{4 + 9}}{{12}}}}{{\dfrac{{12 - 3}}{{12}}}}} \right] - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0 \\
  {\tan ^{ - 1}}\left[ {\dfrac{{\dfrac{{13}}{{12}}}}{{\dfrac{9}{{12}}}}} \right] - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0 \\
  {\tan ^{ - 1}}\left[ {\dfrac{{13}}{{12}} \times \dfrac{{12}}{9}} \right] - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0
 \]
On further simplification, we will get
\[
  {\tan ^{ - 1}}\left[ {\dfrac{{13}}{9}} \right] - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0 \\
  {\tan ^{ - 1}}\left[ {\dfrac{{13}}{9}} \right] = {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) \\
  \tan \left( {{{\tan }^{ - 1}}\left[ {\dfrac{{13}}{9}} \right]} \right) = \tan \left( {{{\tan }^{ - 1}}\left( {\dfrac{x}{3}} \right)} \right) \\
  \dfrac{{13}}{9} = \dfrac{x}{3}
 \]
Furthermore simplification, we will get
\[
  9 \times x = 13 \times 3 \\
  9x = 39 \\
  x = \dfrac{{39}}{9} \\
  x = \dfrac{{13}}{3}
 \]
Therefore, If \[{\tan ^{ - 1}}\left( {\dfrac{1}{3}} \right) + {\tan ^{ - 1}}\left( {\dfrac{3}{4}} \right) - {\tan ^{ - 1}}\left( {\dfrac{x}{3}} \right) = 0\], then \[x\] is equal to \[\dfrac{{13}}{3}\].
Hence, option (D) is correct answer

Additional information: Inverse trigonometric functions are simply the inverse functions of the fundamental trigonometric function sine, cosine, tangent, cotangent, secant, and cosecant. These functions are also known as arc functions. These trigonometry inverse functions are used to calculate the angle with any of the trigonometry ratios.

Note: In this given question, the student should first understand what is being asked in the question and then proceed in the right direction to quickly obtain the correct answer. Furthermore, we should remember the formula \[{\tan ^{ - 1}}A + {\tan ^{ - 1}}B = {\tan ^{ - 1}}\left( {\dfrac{{A + B}}{{1 - AB}}} \right)\] and use it in the given questions to solve them quickly and without any difficult calculations. And, even though the question is simple, we should avoid making calculation errors when answering it.