Prove that \(\displaystyle{\left(\sqrt{{{3}}}-{4}{\sin{{\left({\frac{{{2}\pi}}{{{15}}}}\right)}}}\right)}{\cos{{\left({\frac{{\pi}}{{{30}}}}\right)}}}={\sin{{\left({\frac{{\pi}}{{{30}}}}\right)}}}\)

Rowan Callahan

Rowan Callahan

Answered question

2022-04-01

Prove that
(34sin(2π15))cos(π30)=sin(π30)

Answer & Explanation

Cassius Villarreal

Cassius Villarreal

Beginner2022-04-02Added 11 answers

(34sin24)cos6=sin6
will be true
tan60tan6=4sin24
sin(606)=4sin24cos60cos6
sin54=2sin24cos6 as cos60=12
Using Werner Formulas, the Right Hand Side becomes
sin30+sin18
So, we need to show
sin54sin18=12 as sin30=12
Multiplying
S=sin54sin18=sin54+sin(18)
by 2sin(54(18)2)=2sin36
2sin36S=2sin36sin54+2sin36sin(18)
Applying Werner Formula in the Right Hand Side,
2sin36S=cos18cos90+cos54cos18
S=cos542sin36=12

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