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Compute \((1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 45^\circ)\)

 Jul 23, 2020
 #1
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+2

Compute 

\((1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 45^\circ)\)

 

Formula:

\(\begin{array}{|rcll|} \hline \tan(x-y) &=& \dfrac{\tan x-\tan y}{1+\tan x \tan y }\\\\ \tan(45^\circ) &=& 1 \\ \hline \end{array}\)

 

 

\(\small{ \begin{array}{|rcll|} \hline && \mathbf{(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 22^\circ)(1 + \tan 23^\circ) \dotsm (1 + \tan 43^\circ)(1 + \tan 44^\circ)(1 + \tan 45^\circ)} \\\\ &=& (1 + \tan 1^\circ)(1 + \tan 44^\circ) \times (1 + \tan 2^\circ)(1 + \tan 43^\circ) \times (1 + \tan 3^\circ)(1 + \tan 42^\circ) \\ && \vdots \\ && \times (1 + \tan 22^\circ)(1 + \tan 23^\circ) \times (1 + \tan 45^\circ) \\\\ &=& (1 + \tan 1^\circ)\Big(1 + \tan (45^\circ-1^\circ)\Big) \times (1 + \tan 2^\circ)\Big(1 + \tan (45^\circ-2^\circ)\Big) \times (1 + \tan 3^\circ)\Big(1 + \tan (45^\circ-3^\circ)\Big) \\ && \vdots \\ && \times (1 + \tan 22^\circ)\Big(1 + \tan (45^\circ-22^\circ)\Big) \times (1 + \tan 45^\circ) \\\\ &=& (1 + \tan 1^\circ)\Big(1 + \dfrac{\tan 45^\circ-\tan 1^\circ} {1+\tan 45^\circ\tan 1^\circ} \Big) \times (1 + \tan 2^\circ)\Big(1 + \dfrac{\tan 45^\circ-\tan 2^\circ} {1+\tan 45^\circ\tan 2^\circ} \Big) \times (1 + \tan 3^\circ)\Big(1 + \dfrac{\tan 45^\circ-\tan 3^\circ} {1+\tan 45^\circ\tan 3^\circ} \Big) \\ && \vdots \\ && \times (1 + \tan 22^\circ)\Big(1 + \dfrac{\tan 45^\circ-\tan 22^\circ} {1+\tan 45^\circ\tan 22^\circ}\Big) \times (1 + \tan 45^\circ) \quad | \quad \mathbf{\tan 45^\circ = 1} \\\\ &=& (1 + \tan 1^\circ)\Big(1 + \dfrac{1-\tan 1^\circ} {1+\tan 1^\circ} \Big) \times (1 + \tan 2^\circ)\Big(1 + \dfrac{1-\tan 2^\circ} {1+\tan 2^\circ} \Big) \times (1 + \tan 3^\circ)\Big(1 + \dfrac{1-\tan 3^\circ} {1+\tan 3^\circ} \Big) \\ && \vdots \\ && \times (1 + \tan 22^\circ)\Big(1 + \dfrac{1-\tan 22^\circ} {1+\tan 22^\circ}\Big) \times (1 + 1) \\\\ &=& (1 + \tan 1^\circ)*\dfrac{(1+\tan 1^\circ+1-\tan 1^\circ)} {(1+\tan 1^\circ)} \times (1 + \tan 2^\circ)*\dfrac{(1+\tan 2^\circ+1-\tan 2^\circ)} {(1+\tan 2^\circ)} \times (1 + \tan 3^\circ)*\dfrac{(1+\tan 3^\circ+1-\tan 3^\circ)} {(1+\tan 3^\circ)} \\ && \vdots \\ && \times (1 + \tan 22^\circ)*\dfrac{(1+\tan 22^\circ+1-\tan 22^\circ)} {(1+\tan 22^\circ)} \times 2 \\\\ &=& 2\times 2\times 2 \\ && \vdots \\ && \times 2 \times 2 \\\\ &=& 2^{22}\times 2 \\\\ &=& 2^{23} \\\\ &=& \mathbf{8388608} \\ \hline \end{array} }\)

 

laugh

 Jul 23, 2020
edited by heureka  Jul 23, 2020

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