How can I find the following product using elementary trigonometry?
$$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$
I have tried using a substitution, but nothing has worked.
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First, let's re-arrange these terms so that we can make use of the hints in other answers.
$$\tan(1^\circ) \cdot \tan(89^\circ) \cdot \tan(2^\circ) \cdot \tan(88^\circ) \cdot\cdot\cdot \tan(44^\circ) \cdot \tan(46^\circ) \cdot \tan(45^\circ)$$
Here, we can see a clear pattern of $$\tan(x) \cdot\tan(90^\circ-x)$$ repeating, except for 45, who has no dance partner.
Now, we can use the fact that $$\tan(x) \cdot \tan(90^\circ - x) = 1$$ and reduce all the pairs of numbers to 1. We're left with $$ 1 \cdot 1 \cdot 1 \cdot \cdot \cdot \tan(45^\circ)$$
and since $$ tan(45^\circ) = 1 $$
we get an answer of $$\tan(1^\circ) \cdot \tan(2^\circ) \cdot \tan(3^\circ) \cdot\cdot\cdot \tan(89^\circ) = 1$$