JEE MAIN - Mathematics (2021 - 26th August Evening Shift - No. 11)
The value of
$$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} \right)$$ is :
$$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} \right)$$ is :
$${1 \over {4\sqrt 2 }}$$
$${1 \over 4}$$
$${1 \over 8}$$
$${1 \over {8\sqrt 2 }}$$
Explanation
$$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} \right)$$
$$2{\sin ^2}{\pi \over 8}{\sin ^2}{{2\pi } \over 8}{\sin ^2}{{3\pi } \over 8}$$
$${\sin ^2}{\pi \over 8}{\sin ^2}{{3\pi } \over 8}$$
$${\sin ^2}{\pi \over 8}{\cos ^2}{\pi \over 8}$$
$${1 \over 4}{\sin ^2}\left( {{\pi \over 4}} \right) = {1 \over 8}$$
$$2{\sin ^2}{\pi \over 8}{\sin ^2}{{2\pi } \over 8}{\sin ^2}{{3\pi } \over 8}$$
$${\sin ^2}{\pi \over 8}{\sin ^2}{{3\pi } \over 8}$$
$${\sin ^2}{\pi \over 8}{\cos ^2}{\pi \over 8}$$
$${1 \over 4}{\sin ^2}\left( {{\pi \over 4}} \right) = {1 \over 8}$$
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