QJEEM22Ma9c72: JEE Main 2022 Mathematics - Class 11 JEEM22Ma9c72 Solution: If , then is equal to __________.

$(\sin 10^\circ \,.\,\sin 50^\circ \,.\,\sin 70^\circ )\,.\,(\sin 10^\circ \,.\,\sin 20^\circ \,.\,\sin 40^\circ )$

$= \left( {{1 \over 4}\sin 30^\circ } \right)\,.\,\left[ {{1 \over 2}\sin 10^\circ (\cos 20^\circ - \cos 60^\circ )} \right]$

$= {1 \over {16}}\left[ {\sin 10^\circ \left( {\cos 20^\circ - {1 \over 2}} \right)} \right]$

$= {1 \over {32}}[2\sin 10^\circ \,.\,\cos 20^\circ - \sin 10^\circ ]$

$= {1 \over {32}}[\sin 30^\circ - \sin 10^\circ - \sin 10^\circ ]$

$= {1 \over {64}} - {1 \over {64}}\sin 10^\circ$

Clearly, $\alpha = {1 \over {64}}$

Hence $16 + {\alpha ^{ - 1}} = 80$