Ahh! Too hot, my skin can tan

Geometry Level 2

tan 1 tan 2 tan 3 tan 8 9 = ? \large \tan1^\circ \tan 2^\circ \tan 3^\circ \cdots \tan89^\circ = \, ?


The answer is 1.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

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

tan 1 tan 2 . . . tan 45 cot ( 90 46 ) cot ( 90 47 ) . . . cot ( 90 89 ) \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 tan x = 1 \cot { x } \cdot \tan { x } =1 , then we can pair the terms as follows

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

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...