Question

Find the value of $4{{\tan }^{-1}}\left( \dfrac{1}{5} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right)+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)=$
1. $\dfrac{\pi }{2}$
2. $\dfrac{\pi }{3}$
3. $\dfrac{\pi }{4}$
4. None of these

Answer 1

To find the value of the given expression we will use the trigonometric formula related to the tangent function. We will use following formulas in order to solve the given expression
$2{{\tan }^{-1}}x={{\tan }^{-1}}\dfrac{2x}{1-{{x}^{2}}}$
${{\tan }^{-1}}x-{{\tan }^{-1}}y={{\tan }^{-1}}\dfrac{x-y}{1+xy}$

Complete step by step answer:
We have been given an expression $4{{\tan }^{-1}}\left( \dfrac{1}{5} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right)+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)$.
We have to find the value of the given expression.
First we will rearrange the terms of the given expression. Then we will get
$\Rightarrow 2\left[ 2{{\tan }^{-1}}\left( \dfrac{1}{5} \right) \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right)$
Now, we know that $2{{\tan }^{-1}}x={{\tan }^{-1}}\dfrac{2x}{1-{{x}^{2}}}$.
Now, applying the formula to the above obtained equation we will get
$\Rightarrow 2\left[ {{\tan }^{-1}}\dfrac{2\left( \dfrac{1}{5} \right)}{1-{{\left( \dfrac{1}{5} \right)}^{2}}} \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right)$
Now, simplifying the above obtained equation we will get

& \Rightarrow 2\left[ {{\tan }^{-1}}\dfrac{\left( \dfrac{2}{5} \right)}{\left( \dfrac{25-1}{25} \right)} \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right) \\\ & \Rightarrow 2\left[ {{\tan }^{-1}}\dfrac{\left( \dfrac{2}{5} \right)}{\left( \dfrac{24}{25} \right)} \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right) \\\ & \Rightarrow 2\left[ {{\tan }^{-1}}\dfrac{5}{12} \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right) \\\ \end{aligned}$$ Now, again applying the formula $2{{\tan }^{-1}}x={{\tan }^{-1}}\dfrac{2x}{1-{{x}^{2}}}$ we will get $$\Rightarrow \left[ {{\tan }^{-1}}\dfrac{2\left( \dfrac{5}{12} \right)}{1-{{\left( \dfrac{5}{12} \right)}^{2}}} \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right)$$ Now, simplifying the above obtained equation we will get $$\begin{aligned} & \Rightarrow \left[ {{\tan }^{-1}}\dfrac{\left( \dfrac{5}{6} \right)}{\left( \dfrac{144-25}{144} \right)} \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right) \\\ & \Rightarrow \left[ {{\tan }^{-1}}\dfrac{\left( \dfrac{5}{6} \right)}{\left( \dfrac{119}{144} \right)} \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right) \\\ & \Rightarrow \left[ {{\tan }^{-1}}\dfrac{120}{119} \right]+{{\tan }^{-1}}\left( \dfrac{1}{99} \right)-{{\tan }^{-1}}\left( \dfrac{1}{70} \right) \\\ \end{aligned}$$ Now, we know that ${{\tan }^{-1}}x-{{\tan }^{-1}}y={{\tan }^{-1}}\dfrac{x-y}{1+xy}$ Now, applying the above formula to the obtained equation we will get $$\Rightarrow \left[ {{\tan }^{-1}}\dfrac{120}{119} \right]+{{\tan }^{-1}}\left( \dfrac{\dfrac{1}{99}-\dfrac{1}{70}}{1+\dfrac{1}{99}\times \dfrac{1}{70}} \right)$$ Now, simplifying the above obtained equation we will get $$\begin{aligned} & \Rightarrow \left[ {{\tan }^{-1}}\dfrac{120}{119} \right]+{{\tan }^{-1}}\left( \dfrac{\dfrac{70-99}{6930}}{1+\dfrac{1}{6930}} \right) \\\ & \Rightarrow \left[ {{\tan }^{-1}}\dfrac{120}{119} \right]+{{\tan }^{-1}}\left( \dfrac{\dfrac{-29}{6930}}{\dfrac{6930+1}{6930}} \right) \\\ & \Rightarrow \left[ {{\tan }^{-1}}\dfrac{120}{119} \right]+{{\tan }^{-1}}\left( \dfrac{-29}{6931} \right) \\\ & \Rightarrow {{\tan }^{-1}}\left( \dfrac{120}{119} \right)-{{\tan }^{-1}}\left( \dfrac{29}{6931} \right) \\\ \end{aligned}$$ Now, again applying the formula ${{\tan }^{-1}}x-{{\tan }^{-1}}y={{\tan }^{-1}}\dfrac{x-y}{1+xy}$ to the above obtained equation we will get $$\Rightarrow {{\tan }^{-1}}\left( \dfrac{\dfrac{120}{119}-\dfrac{29}{6931}}{1+\dfrac{120}{119}\times \dfrac{29}{6931}} \right)$$ Now, simplifying the above obtained equation we will get $$\begin{aligned} & \Rightarrow {{\tan }^{-1}}\left( \dfrac{\dfrac{120\times 6931-29\times 119}{119\times 6931}}{1+\dfrac{120\times 29}{119\times 6931}} \right) \\\ & \Rightarrow {{\tan }^{-1}}\left( \dfrac{831720-3451}{824809+3480} \right) \\\ & \Rightarrow {{\tan }^{-1}}\left( \dfrac{828269}{828269} \right) \\\ & \Rightarrow {{\tan }^{-1}}(1) \\\ & \Rightarrow \dfrac{\pi }{4} \\\ \end{aligned}$$ Hence above is the required value of the given expression. **So, the correct answer is “Option 3”.** **Note:** To solve such types of questions students must know the basic concepts of trigonometry. As the calculation is lengthy in this particular question so please avoid calculation mistakes. We can also simplify the numbers by multiplying or dividing. We can simplify the numbers so that the calculation becomes easy.