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Write the value of $\tan \left( 2{{\tan }^{-1}}\dfrac{1}{5} \right)$.

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Last updated date: 14th Jul 2024
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Answer
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Hint: We will find the value of the given function using multiple angle formula for tangent function. We have to start solving by applying the double angle formula of the tangent, $2{{\tan }^{-1}}x={{\tan }^{-1}}\left( \dfrac{2x}{1-{{x}^{2}}} \right)$ and then the trigonometric identity $\tan ({{\tan }^{-1}}x)=x$ can be used to get the final answer.

Complete step-by-step solution:
We have to find the value of $\tan \left( 2{{\tan }^{-1}}\dfrac{1}{5} \right)$.
We will go step by step.
For that, first we have to find the value of $\left( 2{{\tan }^{-1}}\dfrac{1}{5} \right)$.
For finding value we will use trigonometric multiple angle formula. The trigonometric functions of the multiple angles are multiple angle formulas.
Sine, cosine, and tangent are general functions for the multiple angle formula.
Double and triple angle formula is under the trigonometric multiple angle formula.
So, here we will apply the trigonometric multiple angle formula for the tangent.
One of trigonometric multiple angle formula for tangent, i.e the double angle formula is stated as,
$2{{\tan }^{-1}}x={{\tan }^{-1}}\left( \dfrac{2x}{1-{{x}^{2}}} \right)$
Here, we have $x=\dfrac{1}{5}$ . So, we will substitute for x, we will get
$2{{\tan }^{-1}}\dfrac{1}{5}={{\tan }^{-1}}\left( \dfrac{2\left( \dfrac{1}{5} \right)}{1-{{\left( \dfrac{1}{5} \right)}^{2}}} \right)$
Now, simplifying further, we get
\[\begin{align}
  & \Rightarrow 2{{\tan }^{-1}}\dfrac{1}{5}={{\tan }^{-1}}\left( \dfrac{\dfrac{2}{5}}{1-\dfrac{1}{25}} \right) \\
 & \Rightarrow 2{{\tan }^{-1}}\dfrac{1}{5}={{\tan }^{-1}}\left( \dfrac{\dfrac{2}{5}}{\dfrac{25-1}{25}} \right) \\
 & \Rightarrow 2{{\tan }^{-1}}\dfrac{1}{5}={{\tan }^{-1}}\left( \dfrac{\dfrac{2}{5}}{\dfrac{24}{25}} \right) \\
 & \Rightarrow 2{{\tan }^{-1}}\dfrac{1}{5}={{\tan }^{-1}}\left( \dfrac{2}{5}\times \dfrac{25}{24} \right) \\
 & \therefore 2{{\tan }^{-1}}\dfrac{1}{5}={{\tan }^{-1}}\left( \dfrac{5}{12} \right) \\
\end{align}\]
So, now we have computed the value of $2{{\tan }^{-1}}\dfrac{1}{5}$.
Now we will substitute it in give expression, we get,
\[\begin{align}
  & \tan \left( 2{{\tan }^{-1}}\dfrac{1}{5} \right)=\tan \left( {{\tan }^{-1}}\left( \dfrac{5}{12} \right) \right) \\
 & \\
\end{align}\]
Now we have to recall trigonometric identity here.
Here, tan and arc tan are opposite operations. So, they cancel each other out.
That means $\tan ({{\tan }^{-1}}x)=x$
So, we will have \[\begin{align}
  & \tan \left( {{\tan }^{-1}}\left( \dfrac{5}{12} \right) \right)=\dfrac{5}{12} \\
 & \\
\end{align}\]
We get the required answer as \[\dfrac{5}{12}\].

Note: Double and triple angle formula is under multiple angle formula. Here we are using double angle formula for the tangent. Do not find it directly using the calculator you have to remember trigonometric identities for this type of questions. Here students must not convert the tan function in terms of sine and cosine functions as we already have a formula for $2{{\tan }^{-1}}x$. If students convert in terms of sine and cosine, it might get complicated and require the usage of many formulas and identities.