Question

If $\tan A = \dfrac{{ - 1}}{2}$ and $\tan B = \dfrac{{ - 1}}{3}$ then $A + B = $
A. $\dfrac{\pi }{4}$
B. $\dfrac{{3\pi }}{4}$
C. $\dfrac{{5\pi }}{4}$
D. None

Answer 1

Hint: In order to solve this type of question, we will use trigonometric identity $\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$ . We will substitute the given values of \[\tan A\] and \[\tan B\]in the above chosen trigonometric identity. Then we will solve it by simplifying it in order to get the desired correct answer.

Formula used:
$\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$

Complete step by step solution:
We are given that,
$\tan A = \dfrac{{ - 1}}{2}$ ………………..equation$\left( 1 \right)$
$\tan B = \dfrac{{ - 1}}{3}$ ………………..equation$\left( 2 \right)$
We know that,
$\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$
Substituting equation $\left( 1 \right)$ and $\left( 2 \right)$ in the above equation,
$\tan \left( {A + B} \right) = \left[ {\dfrac{{\left( {\dfrac{{ - 1}}{2}} \right) + \left( {\dfrac{{ - 1}}{3}} \right)}}{{1 - \left( {\dfrac{{ - 1}}{2}} \right)\left( {\dfrac{{ - 1}}{3}} \right)}}} \right]$
$\tan \left( {A + B} \right) = \left[ {\dfrac{{\left( {\dfrac{{ - 3 - 2}}{6}} \right)}}{{1 - \left( {\dfrac{1}{6}} \right)}}} \right]$
Simplifying it,
$\tan \left( {A + B} \right) = \left[ {\dfrac{{\left( {\dfrac{{ - 5}}{6}} \right)}}{{\left( {\dfrac{5}{6}} \right)}}} \right]$
$\tan \left( {A + B} \right) = \left( { - 1} \right)$
Solving it further,
$A + B = {\tan ^{ - 1}}\left( { - 1} \right)$
$A + B = \dfrac{{3\pi }}{4}$ $\left[ {\because {{\tan }^{ - 1}}\left( { - 1} \right) = \dfrac{{3\pi }}{4}} \right]$

$\therefore $ The correct option is B.

Note: We have to choose the correct trigonometric identities as these types of questions require the use of correct application of trigonometric rules to get the correct answer.As we have been given the values of \[\tan A\] and \[\tan B\] we have selected the identity of the tangent. Apart from the trigonometric identities we must also remember all the values of the trigonometric functions at each of the specific angles.