The value of tan A + 2 tan 2A + 4 tan 4A + 8 cot 8A is

Question

Smart Achievers · Accepted Answer

(1)

$\tan A + 2 \tan 2 A + 4 \tan 4 A + 8 (\frac{1 - \tan^{2} 4 A}{2 \tan 4 A})$

$= \tan A + 2 \tan 2 A + (\frac{4 \tan^{2} 4 A + 4 - 4 \tan^{2} 4 A}{\tan 4 A})$

$= \tan A + 2 \tan 2 A + 4 \cot 4 A$

$= \tan A + 2 \tan 2 A + 4 (\frac{1 - \tan^{2} 2 A}{2 \tan 2 A})$

$= \tan A + [\frac{2 \tan^{2} 2 A + 2 - 2 \tan^{2} 2 A}{\tan 2 A}]$

$= \tan A + 2 \cot 2 A$

$= \tan A + 2 (\frac{1 - \tan^{2} A}{2 \tan A})$

$= \frac{\tan^{2} A + 1 - \tan^{2} A}{\tan A} = \cot A$

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