Question

cos10cos30cos50cos70 = 3/16

Answer 1

@phi

Answer 2

looks tricky. any ideas ?

Answer 3

no

Answer 4

\[\cos10^{o} \cos30^{o} \cos50^{o} \cos70^{o} = \frac{ 3 }{ 16}\]

Answer 5

\[\cos 30\cos 70 \cos 10 \cos 50=\frac{ \sqrt{3} }{ 2 }*\frac{ 1 }{ 2 }\left( 2\cos 70 \cos 10 \right)\cos 50\]
\[=\frac{ \sqrt{3} }{ 4 }\left( \cos \left( 70-10 \right)+\cos \left( 70+10 \right) \right)\cos 50\]
\[=\frac{ \sqrt{3} }{ 4 }\left( \cos 60+\cos 80 \right)\cos 50\]
\[=\frac{ \sqrt{3} }{ 4 }\left( \frac{ 1 }{ 2 }+\cos 80 \right)\cos 50\]
can you proceed further?

Answer 6

\[=\frac{ \sqrt{3} }{ 8 }\left( \cos 50+(2\cos 80 \cos 50 \right))=?\]

Answer 7

\[=\frac{ \sqrt{3} }{ 8 }\left[ \cos 50+\cos \left( 80-50 \right)+\cos \left( 80+50 \right) \right]\]
\[=\frac{ \sqrt{3} }{ 8 }\left[ \cos 50+\cos 30+\cos 130 \right]\]
\[=\frac{ \sqrt{3} }{ 8 }\left[ \cos 50+\frac{ \sqrt{3} }{ 2 }+\cos \left( 180-50 \right) \right]\]
\[=\frac{ \sqrt{3} }{ 8 }\left[ \cos 50+\frac{ \sqrt{3} }{ 2 }-\cos 50 \right]=?\]