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Villaretq0

Villaretq0

Answered question

2022-06-22

Proving that sin ( 54 ° ) sin ( 66 ° ) = sin ( 48 ° ) sin ( 96 ° )

Answer & Explanation

rioolpijpgp

rioolpijpgp

Beginner2022-06-23Added 19 answers

The LHS is
sin ( 54 ) sin ( 66 ) = cos ( 36 ) cos ( 24 ) .
The RHS is
sin ( 48 ) sin ( 96 ) = 2 sin ( 24 ) cos ( 24 ) sin ( 96 ) = cos ( 24 ) ( cos ( 72 ) cos ( 120 ) ) = = cos ( 24 ) ( cos ( 72 ) + 1 2 ) .
Thus, proving that LHS=RHS is equivalent to proving that
cos ( 36 ) cos ( 72 ) = 1 2 .
The cosine values in the last identity are related to the golden section via the regular pentagon as
cos ( 36 ) = 5 + 1 4 , cos ( 72 ) = 5 1 4 ,
which makes the red identity true.

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