Question

sin 10 *sin 30 *sin 50 *sin 70 *=1/16 please solve it

Answer 1

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Answer 2

Note: Here I am writing theta as A

Given Equation is sin10 sin30 sin50 sin70.

= > It can be written as sin(90 - 80) * sin (90 - 60) * sin(90 - 40) * sin(90 - 20)

We know that sin(90 - A) = cos A

= > cos 80 * cos 60 * cos 40 * cos 20

We know that $cos A = \frac{sin2 A}{2sinA}$

= > $\frac{sin2(80)}{2sin(80)} * \frac{sin2(60)}{2sin(60)} * \frac{sin2(40)}{2sin(40)} * \frac{sin2(20)}{2sin(20)}$

$= \ \textgreater \ \frac{sin160}{2sin80} * \frac{sin120}{2sin60} * \frac{sin80}{2sin40} * \frac{sin40}{2sin20}$


$= \ \textgreater \ \frac{sin160}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2sin20}$


$= \ \textgreater \ \frac{sin160}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2sin(180 - 160)}$


We know that sin(180 - A) = sinA


$= \ \textgreater \ \frac{sin160}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2sin160}$


= > $\frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2}$


= > $\frac{1}{16}$




Hope this helps!