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Let $2A + B$ and $A + 2B$ be acute angles such that $\sin(2A + B) = \frac{\sqrt{3}}{2}$ and $\tan(A + 2B) = 1$. Find the value of $\cot(4A - 7B)$.
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$\sin(2A + B) = \frac{\sqrt{3}}{2} \implies 2A + B = 60^\circ$ --- (1)
$\tan(A + 2B) = 1 \implies A + 2B = 45^\circ$ --- (2)
Solving (1) $\&$ (2), we get $A = 25^\circ$ and $B = 10^\circ$
$\cot(4A - 7B) = \cot 30^\circ = \sqrt{3}$
$\tan(A + 2B) = 1 \implies A + 2B = 45^\circ$ --- (2)
Solving (1) $\&$ (2), we get $A = 25^\circ$ and $B = 10^\circ$
$\cot(4A - 7B) = \cot 30^\circ = \sqrt{3}$