Question

If P = sin a .sin 2a . sin 3a . . . . . . sin 999a &

Q = sin 2a .sin 4a . sin 8a . . . . . . sin 1998a

Find the value of $\frac{Q}{P}$ Where a = $\frac{2\pi }{1999}$

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Answer 1

We have

Q = sin 2a .sin 4a . sin 8a . . . . . . sin 1998a

If we observe both the expression P & Q. we find the angles given in Q just double as angles given in P.

In Q trigonometric functions from sin 1000a can be reduce to lower values.

sin1000a = -sin(2$\pi$ - 1000a) = -sin $(2\pi -1000\times \frac{2\pi }{1999})$

$=-sin\left\{2\pi (1-\frac{1000}{1999})\right\}=-sin\left\{\frac{2\pi \times 999}{1999}\right\}$

= -sin 999a

Similarly

sin 1002a = -sin $(2\pi -1002a)$ = -sin $(2\pi -1002a\times \frac{2\pi }{1999})$

= - sin $\left(2\pi (1-\frac{1002}{1999})\right)$

= -sin$\left(\frac{2\pi \times 997}{1999}\right)$

= -sin 997a

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sin 1004a = -sin 995a

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sin 1998a = -sin a

Substituting all the values in expression Q

Q = (sin 2a . sin 4a . sin 8a . . . . . . sin 998a) $\times$ (- sin 999a) (- sin 997) $\times$ (-sin 995a) . . . . . . (- sin a)

(-) is 998 times or even number of times.

So, it becomes positive

Q = sin 2a .sin 4a . sin 8a . . . . . . sin 998a .sin 999a.sin 997a . sin 995a . . . . .sin 3a . sin a

Q = sin a .sin 2a . sin 3a . sin 4a . . . . . sin 999a

So, Q = P

$\frac{Q}{P}=1$