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What is the value of $2\tan^{-1} \left(\frac{1}{2}\right)$

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Evaluate integers raised to rational exponents

Full solution

Q. What is the value of

2\tan^{-1} \left(\frac{1}{2}\right)

Identify expression: Identify the expression to simplify. $\newline$ We need to find the value of $2\tan^{-1}(\frac{1}{2})$ .
Apply double angle formula: Apply the formula for the double angle of tangent. $\newline$ Using the identity for double angles, $\tan(2a) = \frac{2\tan(a)}{(1 - \tan^2(a))}$ , where $a = \tan^{-1}(\frac{1}{2})$ .
Calculate $\tan^{-1}(\frac{1}{2}):$ Calculate $\tan^{-1}(\frac{1}{2})$ . $\newline$ Let $a = \tan^{-1}(\frac{1}{2})$ , then $\tan(a) = \frac{1}{2}$ .
Substitute into formula: Substitute $\tan(a)$ into the double angle formula. $\tan(2a) = \frac{2\times\left(\frac{1}{2}\right)}{1 - \left(\frac{1}{2}\right)^2}$ $= \frac{1}{1 - \frac{1}{4}}$ $= \frac{1}{\frac{3}{4}}$ $= \frac{4}{3}$ .

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