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Question Number 19291 by Tinkutara last updated on 08/Aug/17

Prove that (1/(cos 6°)) + (1/(sin 24°)) + (1/(sin 48°)) = (1/(sin 12°)) .

Provethat1cos6°+1sin24°+1sin48°=1sin12°.

Answered by Tinkutara last updated on 16/Aug/17

(1/(cos 6°)) − (1/(sin 12°)) + (1/(sin 24°)) + (1/(sin 48°)) = (1/(cos 6°)) − (1/(sin 12°)) + ((sin 48° + sin 24°)/(sin 48° sin 24°)) = (1/(cos 6°)) − (1/(sin 12°)) + ((sin 36°)/(sin 48° sin 12°)) = (1/(cos 6°)) − (1/(sin 12°))(((sin 48° − sin 36°)/(sin 48°))) = (1/(cos 6°)) − (1/(sin 12°))(((2 cos 42° sin 6°)/(sin 48°))) = (1/(cos 6°)) − ((2 sin 6°)/(2 sin 6° cos 6°)) [∵ cos 42° = sin 48°] = (1/(cos 6°)) − (1/(cos 6°)) = 0

1cos6°−1sin12°+1sin24°+1sin48°=1cos6°−1sin12°+sin48°+sin24°sin48°sin24°=1cos6°−1sin12°+sin36°sin48°sin12°=1cos6°−1sin12°(sin48°−sin36°sin48°)=1cos6°−1sin12°(2cos42°sin6°sin48°)=1cos6°−2sin6°2sin6°cos6°[∵cos42°=sin48°]=1cos6°−1cos6°=0

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