Question

Prove that : 8 sin 20 sin 40 sin 80 = (3)^1/2

Answer 1

$\textsf{Given:}$

$\textsf{8\;sin20\;sin40\;sin80}$

$=\textsf{8\;sin40\;sin20\;sin80}$

$=\textsf{8\;sin(60-20)\;sin20\;sin(60+20)}$

$\textsf{Using}$

$\boxed{\mathsf{sin(60-A)\;sinA\;sin(60+A)=\frac{1}{4}sin\,3A}}$

$\textsf{we get}$

$\mathsf{=8[\frac{1}{4}sin\,3(20)]}$

$\mathsf{=8[\frac{1}{4}sin\,60]}$

$\mathsf{=8[\frac{1}{4}\frac{\sqrt3}{2}]}$

$\mathsf{=8[\frac{\sqrt3}{8}]}$

$\mathsf{=\sqrt3}$

$\implies\,\boxed{\mathsf{8\;sin20\;sin40\;sin80=\sqrt3}}$

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Answer 2

Answer:

this is my answer

...Sayan Banerjee