What is the value of \(\displaystyle{{\csc}^{{2}}{\left({\frac{{\pi}}{{{7}}}}\right)}}+{{\csc}^{{2}}{\left({\frac{{{2}\pi}}{{{7}}}}\right)}}+{{\csc}^{{2}}{\left({\frac{{{4}\pi}}{{{7}}}}\right)}}\)

Salvatore Dorsey

Salvatore Dorsey

Answered question

2022-03-13

What is the value of
csc2(π7)+csc2(2π7)+csc2(4π7)

Answer & Explanation

bridged4xi

bridged4xi

Beginner2022-03-14Added 2 answers

If 7x=π, 4x=π3x
sin4x=sin(π3x)=sin3x
2sin2xcos2x=3sinx4sin3x
4sinxcosxcos2x=3sinx4sin3x
If sinx0, we have
4cosxcos2x=34sin2x4cosx(12sin2x)=34sin2x
On squaring & rearrangement, 64(sin2x)3112(sin2x)2+56sin2x7=0
which is a cubic equation in sin2x with roots being sin2rπ7
where r=(1 or 6), (2 or 5) and (3 or 4) as sin(7r)π7=sin((7r)π7=sin(πrπ7)=sinrπ7
PSKUsing Vieta's Formula we have,
sin2π7sin22π7sin24π7=764 and
sin2π7sin22π7+sin22π7sin24π7+sin2π7=5664
We need to find
=sin2π7sin22π7+sin22π7sin24π7+sin24π7sin2π7sin2π7sin22π7sin24π7=567=8

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