bemath-find-the-value-of-m-1-6-tan-mpi-7-

Question Number 112067 by bemath last updated on 06/Sep/20

\sqrt{b e m a t h - - - -}

f i n d t h e v a l u e o f \underset{m = 1}{\prod^{6}} \tan (\frac{m π}{7}) .

Commented by MJS_new last updated on 06/Sep/20

should be = 0

Commented by bemath last updated on 06/Sep/20

w h y s i r ? n o t h i n g c h o i c e a n s w e r 0

Commented by MJS_new last updated on 06/Sep/20

sorry the sum is 0 but the product is - 7 as

Sir John Santu has shown

Commented by bemath last updated on 06/Sep/20

t h a n k y o u s i r

Answered by john santu last updated on 06/Sep/20

\underset{m = 1}{\prod^{6}} \tan (\frac{m π}{7}) = ?

w e h a v e \sin (n x) = 2^{m - 1} \underset{m = 0}{\prod^{n - 1}} \sin (x + \frac{m π}{n})

w h e n x \to 0, S_{n} = \underset{m = 1}{\prod^{n - 1}} \sin (\frac{m π}{n}) = \frac{n}{2^{n - 1}}

w h e r e n = 7, t h u s S_{7} = \underset{m = 1}{\prod^{7 - 1}} \sin (\frac{m π}{7}) = \frac{7}{64}

b u t w e k n o w r e l a t i o n \cos x = \sin (\frac{π}{2} - x)

\cos (x + \frac{m π}{n}) = - \sin (x - \frac{π}{2} + \frac{m π}{n})

\underset{m = 0}{\prod^{n - 1}} \cos (x + \frac{m π}{n}) = {(- 1)}^{n} \underset{m = 0}{\prod^{n - 1}} \sin (x - \frac{π}{2} + \frac{m π}{n})

= \frac{{(- 1)}^{n}}{2^{n - 1}} . \sin (n x - \frac{n π}{2})

= \frac{{(- 1)}^{n}}{2^{n - 1}} . \frac{\sin (n x - \frac{n π}{2})}{\cos x}

x \to 0; Q_{n} = \underset{m = 1}{\prod^{n - 1}} \cos (\frac{m π}{n}) = \frac{{(- 1)}^{n + 1}}{2^{n - 1}} . \sin (\frac{n π}{2})

p u t n = 7; Q_{7} = \underset{m = 1}{\prod^{7 - 1}} \cos (\frac{m π}{7}) = \frac{{(- 1)}^{8}}{2^{6}} . \sin (\frac{7 π}{2}) = - \frac{1}{64}

t h e r e f o r e \underset{m = 1}{\prod^{6}} \tan (\frac{m π}{7}) = \frac{(\frac{7}{64})}{(- \frac{1}{64})} = - 7

Commented by bemath last updated on 06/Sep/20

t h a n k y o u

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