Tangent product series!

$\tan{\theta} = \cot({90^{\circ}-\theta})$

$\begin{aligned} \text{Product} &=\tan{1^{\circ}} \tan{2^{\circ}} \tan{3^{\circ}} \tan{4^{\circ}} \cdots \tan{86^{\circ}} \tan{87^{\circ}} \tan{88^{\circ}} \tan{89^{\circ}} \\ &=\tan{1^{\circ}} \tan{89^{\circ}} \tan{2^{\circ}} \tan{88^{\circ}} \tan{3^{\circ}} \tan{87^{\circ}} \tan{4^{\circ}} \tan{86^{\circ}} \cdots \\ &=\tan{1^{\circ}} \cot{1^{\circ}} \tan{88^{\circ}} \cot{88^{\circ}} \tan{87^{\circ}} \cot{87^{\circ}} \tan{86^{\circ}} \cot{86^{\circ}} \cdots \\ &=\cancel{\tan{1^{\circ}}} \cancel{\cot{1^{\circ}}} \cancel{\tan{88^{\circ}}} \cancel{\cot{88^{\circ}}} \cancel{\tan{87^{\circ}}} \cancel{\cot{87^{\circ}}} \cancel{\tan{86^{\circ}}} \cancel{\cot{86^{\circ}}} \cdots \\ &=\boxed{1} \end{aligned}$

$\begin{aligned} P &= {\color{#EC7300}{\tan(1^\circ) \times \tan(2^\circ) \ldots \tan(44^\circ)}}\times \blue{\tan(45^\circ)} \times {\color{#69047E}{\tan(46^\circ) \ldots \tan(88^\circ) \times \tan(89^\circ)}} \\ \\ &= {\color{#EC7300}{\tan(1^\circ) \times \tan(2^\circ) \ldots \tan(44^\circ)}}\times \blue{\tan(45^\circ)} \times {\color{#69047E}{\cot(44^\circ) \ldots \cot(2^\circ) \times \cot(1^\circ)}} \\ \\ &= {\color{#EC7300}{\tan(1^\circ)}} \times {\color{#69047E}{\cot(1^\circ)}} \times {\color{#EC7300}{\tan(2^\circ)}} \times {\color{#69047E}{\cot(2^\circ)}} \times {\color{#EC7300}{\tan(3^\circ)}} \times {\color{#69047E}{\cot(3^\circ)}} \ldots \times {\color{#EC7300}{\tan(44^\circ)}} \times {\color{#69047E}{\cot(44^\circ)}} \times \blue{\tan(45^\circ)} \\ \\ &= 1 \times 1 \times 1 \times \ldots \blue{1} \\ \\ P &= \boxed{1} \end{aligned}$