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Solve the following. \[\dfrac{{2\tan 30^\circ }}{{1 - {{\tan }^2}30^\circ }} = \tan \theta \] then \[\theta = \]
A. \[60^\circ \]
B. \[45^\circ \]
C. \[65^\circ \]
D. \[35^\circ \]

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Last updated date: 26th Feb 2024
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IVSAT 2024
Answer
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Hint: Here we will first simplify the given trigonometric equation. Then we will use the basic trigonometric formulas to solve the given trigonometric equation. We will then use mathematical operations like addition, subtraction, multiplication and division. After simplifying the terms, we will get the required value of the angle and hence the required answer.

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

Complete step by step solution:
The given trigonometric equation is \[\dfrac{{2\tan 30^\circ }}{{1 - {{\tan }^2}30^\circ }} = \tan \theta \].
Now, we will use the basic trigonometric formulas for angles to solve the given trigonometric equation.
We know from the basic trigonometric formulas for angles that \[\tan 30^\circ = \dfrac{1}{{\sqrt 3 }}\].
Now, we will substitute this value in the given trigonometric equation.
\[ \Rightarrow \dfrac{{2 \times \dfrac{1}{{\sqrt 3 }}}}{{1 - {{\left( {\dfrac{1}{{\sqrt 3 }}} \right)}^2}}} = \tan \theta \]
On multiplying and squaring the terms, we get
\[ \Rightarrow \dfrac{{\dfrac{2}{{\sqrt 3 }}}}{{1 - \dfrac{1}{3}}} = \tan \theta \]
Now, we will subtract the terms in the denominator. Therefore, we get
\[ \Rightarrow \dfrac{{\dfrac{2}{{\sqrt 3 }}}}{{\dfrac{2}{3}}} = \tan \theta \]
On simplifying the terms, we get
\[ \Rightarrow \dfrac{3}{{\sqrt 3 }} = \tan \theta \]
We can see that the given fraction can be reduced further.
On reducing the fraction, we get
\[ \Rightarrow \sqrt 3 = \tan \theta \]
Now, we will take \[{\tan ^{ - 1}}\] on both sides.
\[ \Rightarrow {\tan ^{ - 1}}\left( {\sqrt 3 } \right) = {\tan ^{ - 1}}\left( {\tan \theta } \right)\]
We know from the inverse trigonometric identities that \[{\tan ^{ - 1}}\left( {\tan \theta } \right) = \theta \].
Using this identity, we get
\[ \Rightarrow {\tan ^{ - 1}}\left( {\sqrt 3 } \right) = \theta \]
We know that the value of \[{\tan ^{ - 1}}\left( {\sqrt 3 } \right)\] is equal to \[60^\circ \].
Now, we will substitute this value here.
\[\begin{array}{l} \Rightarrow 60^\circ = \theta \\ \Rightarrow \theta = 60^\circ \end{array}\]

Hence, the correct option is option A.

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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