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Compute 2*arctan(1/3) + arctan(1/7).

 Dec 3, 2019
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Compute \(2*\arctan\left(\dfrac{1}{3}\right) + \arctan\left(\dfrac{1}{7}\right)\).

 

\(\begin{array}{|rcll|} \hline && 2*\arctan\left(\dfrac{1}{3}\right) + \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan\left(\dfrac{1}{3}\right)+\arctan\left(\dfrac{1}{3}\right) + \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan\left( \dfrac{1}{3} + \dfrac{1}{3} \above 1pt 1-\dfrac{1}{3}\cdot \dfrac{1}{3} \right)+ \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan\left( \dfrac{3}{4} \right)+ \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan\left( \dfrac{3}{4} + \dfrac{1}{7} \above 1pt 1-\dfrac{3}{4}\cdot \dfrac{1}{7} \right)+ \arctan\left(\dfrac{1}{7}\right) \\ &=& \arctan(1) \\ &=& 45^\circ \quad & | \quad \left(\dfrac{\pi}{4}\right) \\ \hline \end{array}\)

 

 

laugh

 Dec 3, 2019

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