Ahh! Too hot, my skin can tan

Geometry Level 2

$\large \tan1^\circ \tan 2^\circ \tan 3^\circ \cdots \tan89^\circ = \, ?$

1 solution

Mateo Matijasevick
May 30, 2016

Knowing that $\tan { x } =\cot { (90-x) }$ , we can rewrite the given expression as

$\tan { 1 } \cdot \tan { 2 } \cdot ...\cdot \tan { 45 } \cdot \cot { (90-46) } \cdot \cot { (90-47) } \cdot ...\cdot \cot { (90-89) }$

As $\cot { x } \cdot \tan { x } =1$ , then we can pair the terms as follows

$\tan { 1 } \cdot \cot { 1 } \cdot \tan { 2 } \cdot \cot { 2 } \cdot ...\cdot \tan { 44 } \cdot \cot { 44 } \cdot \tan { 45 }$

Which is equal to $1\cdot 1\cdot ...\cdot 1\cdot \tan { 45 } =\tan { 45 } =1$