If-k-1-10-cos-kpi-11-2-n-then-n-

Question Number 127835 by bemath last updated on 02/Jan/21

If \underset{k = 1}{\prod^{10}} \cos (\frac{k π}{11}) = - 2^{n}, then n = ?

Answered by liberty last updated on 02/Jan/21

let Y = \cos x \cos 2 x \cos 3 x \cos 4 x

\cos 5 x \cos 6 x \cos 7 x \cos 8 x

\cos 9 x \cos 10 x; where x = \frac{π}{11}

\Rightarrow 2 Ysin x = \sin 2 x \cos 2 x \cos 3 x \cos 4 x

\cos 5 x \cos 6 x \cos 7 x \cos 8 x

\cos 9 x \cos 10 x

\Rightarrow 2^{2} Ysin x = \sin 4 x \cos 3 x \cos 4 x \cos 5 x

\cos 6 x \cos 7 x \cos 8 x \cos 9 x \cos 10 x

\Rightarrow 2^{3} Ysin x = \sin 8 x \cos 3 x \cos 5 x \cos 6 x

\cos 7 x \cos 8 x \cos 9 x \cos 10 x

\Rightarrow 2^{4} Ysin x = \sin 16 x \cos 3 x \cos 5 x \cos 6 x

\cos 7 x \cos 9 x \cos 10 x

note \sin 16 x = \sin (\frac{16 π}{11}) = \sin (π + \frac{5 π}{11}) = - \sin 5 x

\Rightarrow 2^{4} Ysin x = - \sin 5 x \cos 3 x \cos 5 x \cos 6 x

\cos 7 x \cos 9 x \cos 10 x

\Rightarrow 2^{5} Ysin x = - \sin 10 x \cos 3 x \cos 6 x \cos 7 x \cos 9 x \cos 10 x

\Rightarrow 2^{6} Ysin x = - \sin 20 x \cos 3 x \cos 6 x \cos 7 x \cos 9 x

note \sin 20 x = \sin (π + \frac{9 π}{11}) = - \sin 9 x

\Rightarrow 2^{6} Ysin x = \sin 9 x \cos 3 x \cos 6 x \cos 7 x \cos 9 x

\Rightarrow 2^{7} Ysin x = \sin 18 x \cos 3 x \cos 6 x \cos 7 x

note \sin 18 x = \sin (π + \frac{7 π}{11}) = - \sin 7 x

\Rightarrow 2^{7} Ysin x = - \sin 7 x \cos 3 x \cos 6 x \cos 7 x

\Rightarrow 2^{8} Ysin x = - \sin 14 x \cos 3 x \cos 6 x

note \sin 14 x = \sin (π + \frac{3 π}{11}) = - \sin 3 x

\Rightarrow 2^{9} Ysin x = \sin 6 x \cos 6 x

\Rightarrow 2^{10} Ysin x = \sin 12 x

∴ Y = \frac{\sin (π + \frac{π}{11})}{2^{10} \sin (\frac{π}{11})} = \frac{- \sin (\frac{π}{11})}{2^{10} \sin (\frac{π}{11})}

Y = - 2^{- 10}, we get n = - 10 .

Commented by bemath last updated on 02/Jan/21

greatt \dots

Commented by bemath last updated on 02/Jan/21

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