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Given that \[\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A \cdot \tan B}}\] where \[A\] and \[B\] are acute angles. Calculate \[A + B\] when \[\tan A = \dfrac{1}{2}\], \[\tan B = \dfrac{1}{3}\].
A. \[A + B = 30^\circ \]
B. \[A + B = 45^\circ \]
C. \[A + B = 60^\circ \]
D. \[A + B = 75^\circ \]

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Last updated date: 03rd Mar 2024
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IVSAT 2024
Answer
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Hint: Here we will first use the given trigonometric formulas and we will substitute the value of tangent of both the angles in the formula. Then we will simplify it using the mathematical operations like addition, subtraction and multiplication. Then we will use the basics of the inverse trigonometric trigonometry here to get the required sum of the two angles.

Formula used:
Inverse trigonometric identity is given by \[{\tan ^{ - 1}}\left( {\tan \theta } \right) = \theta \].

Complete step by step solution:
Here we need to find the value of the given sum of two acute angles.
It is given that:-
\[\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A \cdot \tan B}}\] ………… \[\left( 1 \right)\]
Now, we will substitute \[\tan A = \dfrac{1}{2}\] and \[\tan B = \dfrac{1}{3}\] in the equation \[\left( 1 \right)\]. Therefore, we get
\[ \Rightarrow \tan \left( {A + B} \right) = \dfrac{{\dfrac{1}{2} + \dfrac{1}{3}}}{{1 - \dfrac{1}{2} \cdot \dfrac{1}{3}}}\]
Simplifying the expression, we get
\[ \Rightarrow \tan \left( {A + B} \right) = \dfrac{{\dfrac{{3 + 2}}{6}}}{{1 - \dfrac{1}{6}}}\]
On adding the terms in the numerator and subtracting the terms in the denominator, we get
\[ \Rightarrow \tan \left( {A + B} \right) = \dfrac{{\dfrac{5}{6}}}{{\dfrac{5}{6}}}\]
On further simplification, we get
\[ \Rightarrow \tan \left( {A + B} \right) = 1\]
Now, we will take \[{\tan ^{ - 1}}\] on both sides.
\[ \Rightarrow {\tan ^{ - 1}}\left( {\tan \left( {A + B} \right)} \right) = {\tan ^{ - 1}}\left( 1 \right)\]
We know from the inverse trigonometric identities that \[{\tan ^{ - 1}}\left( {\tan \theta } \right) = \theta \].
Using this identity, we get
\[ \Rightarrow A + B = {\tan ^{ - 1}}\left( 1 \right)\]
We know that the value of \[{\tan ^{ - 1}}\left( 1 \right)\] is equal to \[45^\circ \].
Now, we will substitute this value in the above equation, so we get
\[ \Rightarrow A + B = 45^\circ \].

Hence, the correct option is option B.

Note:
Trigonometry is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’ and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.
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