The value of 2 sin π 8 sin 2 π 8 sin 3 π 8 sin 5 π 8 sin 6 π 8 sin 7 π 8 is : | TestHub

The value of $2 \sin (\frac{π}{8}) \sin (\frac{2 π}{8}) \sin (\frac{3 π}{8}) \sin (\frac{5 π}{8}) \sin (\frac{6 π}{8}) \sin (\frac{7 π}{8})$ is :

Answer:

Solution:

$\Rightarrow 2 \sin (\frac{π}{8}) \sin (\frac{2 π}{8}) \sin (\frac{3 π}{8}) \sin (\frac{5 π}{8}) \sin (\frac{6 π}{8}) \sin (\frac{7 π}{8})$

$\Rightarrow 2 \sin (\frac{π}{8}) \sin (\frac{2 π}{8}) \sin (\frac{3 π}{8}) \sin (π - \frac{3 π}{8}) \sin (π - \frac{2 π}{8}) \sin (π - \frac{π}{8})$

$\Rightarrow 2 \sin^{2} \frac{π}{8} \sin^{2} (\frac{2 π}{8}) \sin^{2} (\frac{3 π}{8})$

$\Rightarrow 2 \sin^{2} \frac{π}{8} \sin^{2} (\frac{π}{4}) \sin^{2} (\frac{3 π}{8})$

$\Rightarrow \sin^{2} \frac{π}{8} \sin^{2} (\frac{3 π}{8})$

$\Rightarrow \sin^{2} \frac{π}{8} \sin^{2} (\frac{π}{2} - \frac{π}{8})$

$\Rightarrow \sin^{2} \frac{π}{8} \cos^{2} \frac{π}{8}$

$\Rightarrow \frac{1}{4} {(2 \sin \frac{π}{8} \cos \frac{π}{8})}^{2}$

$\frac{1}{4} \sin^{2} \frac{π}{4} = \frac{1}{8}$

Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub