Solve: P=\prod_{k=1}^{45}\csc^2(2k-1)^\circ=m^n

tefoldaavv

tefoldaavv

Answered question

2022-04-22

Solve:
P=k=145csc2(2k1)=mn

Answer & Explanation

amisayq6t

amisayq6t

Beginner2022-04-23Added 18 answers

sin(2n+1)x=(2n+1)sinx++(1)n22nsin2n+1x
If we set 2n+1=45, 244sin45x+45sinxsin45x=0
If we set sin45x=sin45, 45x=360m+45 where m is any integer
x=8m+1 where 0m44
Q=m=044sin(8m+1)=121244
Q2=12288
Clearly [sin2(8m+1);0m44]
=[sin2(2r1),1r45]
as x=1,9,17,25,33,41,49,57,65,73,81,89
97[sin97=sin(18097)=sin83]
105[sin75] and so on
m=228m+1=177sin177=sin(1803)=sin3
m=448m+1=353sin353=sin7

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