Multiply the tan

$\begin{aligned} P & = \tan(1^\circ)\tan(2^\circ)\tan(3^\circ)\tan(4^\circ) \cdots \tan(86^\circ)\tan(87^\circ)\tan(88^\circ)\tan(89^\circ) \\ & = \tan(1^\circ)\cot(1^\circ)\tan(2^\circ)\cot(2^\circ)\tan(3^\circ)\cot(3^\circ)\cdots\tan(44^\circ)\cot(44^\circ) \tan(45^\circ) \\ & = 1 \times 1 \times 1 \times \cdots 1 \times 1 = \boxed 1 \end{aligned}$

$\tan{(1)}\tan{(2)}\tan{(3)}\cdots\tan{(43)}\tan{(44)}\tan{(45)}\tan{(46)}\tan{(47)}\cdots\tan{(87)}\tan{(88)}\tan{(89)}$

Using property: $\tan{(A)}=\cot{(90-A)}$

$\cot{(89)}\cot{(88)}\cot{(87)}\cdots\cot{(47)}\cot{(46)}\tan{(45)}\tan{(46)}\tan{(47)}\cdots\tan{(87)}\tan{(88)}\tan{(89)}$

Rearranging:

$(\cot{(89)}\tan{(89)})(\cot{(88)}\tan{(88)})(\cot{(87)}\tan{(87)})\cdots(\cot{(47)}\tan{(47)})(\cot{(46)}\tan{(46)})\tan{(45)}$

Using property: $\cot{(A)}=\frac{1}{\tan{(A)}}$

$1\times1\times1\times\cdots\times1\times1\times\tan{(45)}=\tan{(45)}=\boxed{1}$

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