Download Problem: Given that cos 36 = 1 4 ( 1 + √ 5 ) , show that (tan 2 18

Problem:
Given that
√ 1
cos 36◦ =
1+ 5 ,
4
show that
tan2 18◦ tan2 54◦
is a rational number.
Solution:
√
1 − cos 2u
3− 5
2
◦
√ .
Using the power-reducing formula tan u =
, we find that that tan 18 =
1 + cos 2u
5+ 5
Next, we note that 54 = 3 · 18 and think to ourselves, “gee wouldn’t it be great to have a
way to easily compute tan 3u?” Since we don’t have one, we derive it using trigonometric
sum and difference formulas:
2
tan u + tan 2u
1 − tan u tan 2u
2 tan u
tan u + 1−tan
2u
tan(u + 2u) =
=
2 tan u
1 − tan u 1−tan
2u
3 tan u − tan3 u
1 − 3 tan2 u
=
Now we are ready to do the fun algebra! For the sake of clarity, we let u = 18◦ .
2
tan u
2
tan 3u
3 tan u − tan3 u
= (tan u)
1 − 3 tan2 u
2
3 tan2 u − tan4 u
=
1 − 3 tan2 u

√ 2 2
√
3−√5
3−√5
3
−
5+ 5
 5+ 5

√
= 

3−√5
1 − 3 5+ 5
2
=
2
1
.
5
Note: In the above solution, several routine algebraic manipulations were skipped. However,
a complete solution would include all of these steps.
1