I want to show that, <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="no

Eliaszowyr1

Eliaszowyr1

Answered question

2022-05-22

I want to show that,
Φ tan 9 ϕ tan 27 sin 2 9 sin 2 27 = 4

Answer & Explanation

bluayu0y

bluayu0y

Beginner2022-05-23Added 11 answers

Noting that
ϕ = 2 cos ( π 5 ) = 2 sin ( 3 π 10 ) , Φ = 2 sin ( π 10 )
from Golden ratio, so one has
Φ tan 9 ϕ tan 27 sin 2 9 sin 2 27 = 2 sin ( π 10 ) tan ( π 20 ) 2 sin ( 3 π 10 ) tan ( 3 π 20 ) sin 2 ( π 20 ) sin 2 ( 3 π 20 ) = 2 2 sin ( π 20 ) cos ( π 20 ) tan ( π 20 ) 2 sin ( 3 π 20 ) cos ( 3 π 20 ) tan ( 3 π 20 ) sin 2 ( π 20 ) sin 2 ( 3 π 20 ) = 4 sin 2 ( π 20 ) sin 2 ( 3 π 20 ) sin 2 ( π 20 ) sin 2 ( 3 π 20 ) = 4.

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