The value of 2sin(π)/8sin(2π)/8sin(3π)/8sin(5π)/8sin(6π)/8sin(7π)/8 is :

Correct answer: (B) 1/8. ⇒2sin(π)/8sin(2π)/8sin(3π)/8sin(5π)/8sin(6π)/8sin(7π)/8 ⇒2sin(π)/8sin(2π)/8sin(3π)/8sinπ-(3π)/8sinπ-(2π)/8sinπ-(π)/8 ⇒2sin^2(π)/8sin^2(2π)/8sin^2(3π)/8 ⇒2sin^2(π)/8sin^2(2π)/8sin^2(3π)/8 ⇒2sin^2(π)/8sin^2(π)/4sin^2(3π)/8 ⇒sin^2(π)/8sin^2(3π)/8 ⇒sin^2(π)/8sin^2(π)/2-(π)/8 ⇒sin^2(π)/8cos^2(π)/8 ⇒1/42sinπ8cosπ8^2 1/4sin^2(π)/4=1/8

Q36 / 90 · JEE Main 2021 (26 Aug Shift 2) · Mathematics

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 :

A
$\frac{1}{4 \sqrt{2}}$
Correct answer
B
$\frac{1}{8}$
C
$\frac{1}{8 \sqrt{2}}$
D
$\frac{1}{4}$

Solution

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

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

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

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

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

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

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

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

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

Hence, the correct answer is option: 'B'.

The value of 2sinπ8sin2π8sin3π8sin5π8sin6π8sin7π8 is :

Solution

Relevant Questions

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 :