Question

सिद्ध कीजिए: 16sin100sin300sin500sin700=1

Answer 1

$\text{I think your question should be:}$

$\text{Prove that:}\displaystyle\;16\;sin10^{\circ}\;sin30^{\circ}\;sin50^{\circ}\;sin70^{\circ}=1$

$\text{Consider,}$

$\displaystyle\;16\;sin10^{\circ}\;sin30^{\circ}\;sin50^{\circ}\;sin70^{\circ}$

$=\displaystyle\;16\;sin10^{\circ}(\frac{1}{2})sin50^{\circ}\;sin70^{\circ}$

$=\displaystyle\;8\;sin10^{\circ}\;sin50^{\circ}\;sin70^{\circ}$

$=\displaystyle\;8\;sin(60^{\circ}-10^{\circ})\;sin10^{\circ}\;sin(60^{\circ}-10^{\circ})$

$\text{Using,}$

$\boxed{\bf\;sin(60^{\circ}-A)\;sinA\;sin(60^{\circ}+A)=\frac{1}{4}sin3A}$

$=\displaystyle\;8[\frac{1}{4}sin3(10^{\circ})]$

$=\displaystyle\;8[\frac{1}{4}sin30^{\circ}]$

$=\displaystyle\;8[\frac{1}{4}(\frac{1}{2})]$

$=\displaystyle\;8[\frac{1}{8}]$

$=\displaystyle\;1$

$\therefore\boxed{\bf\;16\;sin10^{\circ}\;sin30^{\circ}\;sin50^{\circ}\;sin70^{\circ}=1}$

Answer 2

Answer:

\text{I think your question should be:}I think your question should be:

\text{Prove that:}\displaystyle\;16\;sin10^{\circ}\;sin30^{\circ}\;sin50^{\circ}\;sin70^{\circ}=1Prove that:16sin10∘sin30∘sin50∘sin70∘=1

\text{Consider,}Consider,

\displaystyle\;16\;sin10^{\circ}\;sin30^{\circ}\;sin50^{\circ}\;sin70^{\circ}16sin10∘sin30∘sin50∘sin70∘

=\displaystyle\;16\;sin10^{\circ}(\frac{1}{2})sin50^{\circ}\;sin70^{\circ}=16sin10∘(21)sin50∘sin70∘

=\displaystyle\;8\;sin10^{\circ}\;sin50^{\circ}\;sin70^{\circ}=8sin10∘sin50∘sin70∘

=\displaystyle\;8\;sin(60^{\circ}-10^{\circ})\;sin10^{\circ}\;sin(60^{\circ}-10^{\circ})=8sin(60∘−10∘)sin10∘sin(60∘−10∘)

\text{Using,}Using,

\boxed{\bf\;sin(60^{\circ}-A)\;sinA\;sin(60^{\circ}+A)=\frac{1}{4}sin3A}sin(60∘−A)sinAsin(60∘+A)=41sin3A

=\displaystyle\;8[\frac{1}{4}sin3(10^{\circ})]=8[41sin3(10∘)]

=\displaystyle\;8[\frac{1}{4}sin30^{\circ}]=8[41sin30∘]

=\displaystyle\;8[\frac{1}{4}(\frac{1}{2})]=8[41(21)]

=\displaystyle\;8[\frac{1}{8}]=8[81]

=\displaystyle\;1=1

\therefore\boxed{\bf\;16\;sin10^{\circ}\;sin30^{\circ}\;sin50^{\circ}\;sin70^{\circ}=1}∴16sin10∘sin30∘sin50∘sin70∘=1