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The value of $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$ is
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Given, $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$
$\begin{aligned}
&=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 87^{\circ} \tan 88^{\circ} \tan 89^{\circ} \\
&=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan \left(90-3^{\circ}\right) \\
&\tan (90-2)^{\circ} \tan (90-1)^{\circ}
\end{aligned}$
$=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \cot 3^{\circ} \cot 2^{\circ} \cot 1^{\circ}$
$=\tan 1^{\circ} \cot 1^{\circ} \tan 2^{\circ} \cot 2^{\circ} \tan 3^{\circ} \cot 3^{\circ} \ldots \tan 45^{\circ}$
$=$ (1) (l) (l) ... (1)
$=1$
$\begin{aligned}
&=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 87^{\circ} \tan 88^{\circ} \tan 89^{\circ} \\
&=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan \left(90-3^{\circ}\right) \\
&\tan (90-2)^{\circ} \tan (90-1)^{\circ}
\end{aligned}$
$=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \cot 3^{\circ} \cot 2^{\circ} \cot 1^{\circ}$
$=\tan 1^{\circ} \cot 1^{\circ} \tan 2^{\circ} \cot 2^{\circ} \tan 3^{\circ} \cot 3^{\circ} \ldots \tan 45^{\circ}$
$=$ (1) (l) (l) ... (1)
$=1$
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