If \[\tan\alpha = \dfrac{m}{{\left( {m + 1} \right)}}\] and \[\tan\beta = \dfrac{1}{{\left( {2m + 1} \right)}}\]. Then find the value of \[\left( {\alpha + \beta } \right)\].
A. \[\dfrac{\pi }{3}\]
B. \[\dfrac{\pi }{4}\]
C. 0
D. \[\dfrac{\pi }{2}\]

Hint: In the given question, two trigonometric equations of \[\tan\] are given. By substituting the given equations in the tangent addition formula, we will find the value of \[\tan \left( {\alpha + \beta } \right)\]. Then using the value of trigonometric angles, we will find the value of \[\left( {\alpha + \beta } \right)\].
Formula Used:
tangent addition formula: \[\tan \left( {A + B} \right) = \dfrac{{\tan A +\ tan B}}{{1 - \tan A\tan B}}\]
\[\tan \left( {\dfrac{\pi }{4}} \right) = 1\]
Complete step by step solution:
The given trigonometric equations are \[\tan\alpha = \dfrac{m}{{\left( {m + 1} \right)}}\] and \[\tan\beta = \dfrac{1}{{\left( {2m + 1} \right)}}\].
Let’s substitute the given equations in the tangent addition formula.
\[\tan \left( {\alpha + \beta } \right) = \dfrac{{\tan\alpha + \tan\beta }}{{1 - \tan\alpha\ tan\beta }}\]
\[ \Rightarrow \]\[\tan \left( {\alpha + \beta } \right) = \dfrac{{\dfrac{m}{{m + 1}} + \dfrac{1}{{2m + 1}}}}{{1 - \left( {\dfrac{m}{{m + 1}}} \right)\left( {\dfrac{1}{{2m + 1}}} \right)}}\]
\[ \Rightarrow \]\[\tan \left( {\alpha + \beta } \right) = \dfrac{{\dfrac{{m\left( {2m + 1} \right) + \left( {m + 1} \right)}}{{\left( {m + 1} \right)\left( {2m + 1} \right)}}}}{{1 - \dfrac{m}{{\left( {m + 1} \right)\left( {2m + 1} \right)}}}}\]
\[ \Rightarrow \]\[\tan \left( {\alpha + \beta } \right) = \dfrac{{2{m^2} + m + m + 1}}{{\left( {m + 1} \right)\left( {2m + 1} \right) - m}}\]
\[ \Rightarrow \]\[\tan \left( {\alpha + \beta } \right) = \dfrac{{2{m^2} + 2m + 1}}{{2{m^2} + m + 2m + 1 - m}}\]
\[ \Rightarrow \]\[\tan \left( {\alpha + \beta } \right) = \dfrac{{2{m^2} + 2m + 1}}{{2{m^2} + 2m + 1}}\]
\[ \Rightarrow \]\[\tan \left( {\alpha + \beta } \right) = 1\]
\[ \Rightarrow \]\[\tan \left( {\alpha + \beta } \right) = \tan \left( {\dfrac{\pi }{4}} \right)\] [Since \[\tan \left( {\dfrac{\pi }{4}} \right) = 1\]]
\[ \Rightarrow \]\[\alpha + \beta = \dfrac{\pi }{4}\]
Hence the correct option is option B
Note: Students are often confused with the formula \[\tan \left( {A + B} \right) = \dfrac{{\tan A +\tan B}}{{1 - \tan A\tan B}}\] and \[\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 + \tan A\tan B}}\] . But the correct tangent addition formula is \[\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A \tan B}}\].