Question

Prove that sin10×sin50×sin70=1/8

Answer 1

Step-by-step explanation:

We have

$\sin(10) . \sin(50) . \sin(70)$

$= \sin(90 - 80) . \sin(90 - 40) . \sin(90 - 20)$

$= \cos(80) . \cos(40) . \cos(20)$

$= \cos(20) . \cos(2 \times 20). \cos(4 \times 20)$

$= \cos(20) . \cos(2 \times 20). \cos( {2}^{2} \times 20)$

Now, we know that,

$\cos( \alpha ) . \cos(2 \alpha ). \cos( {2}^{2} \alpha ) .... \cos( {2}^{n - 1} \alpha ) = \frac{ \sin( {2}^{n} \alpha ) }{ {2}^{n} \sin( \alpha ) }$

$= \frac{ \sin( {2}^{3} \times 20 ) }{ {2}^{3} \sin(20) }$

$= \frac{ \sin(160) }{8 \sin(20) }$

$= \frac{ \sin(180 - 20) }{8 \sin(20) }$

$= \frac{ \sin(20) }{8 \sin(20) }$

$= \frac{1}{8}$