Product Of Cosines

The following trigonometric formulae are applied:

$\begin{aligned} \sin(A) & = & \cos(90^\circ - A) \\ \sin(2A) & = & 2\sin(A) \cos(A) \\ \sin(3A) & = & 4 \sin(A) \sin(60^\circ - A) \sin(60^\circ + A) \\ \end{aligned}$

We group the terms as such:

$\left ( \cos 10^\circ \times \cos 80^\circ \right ) \times \left ( \cos 20^\circ \times \cos 70^\circ \right ) \times \left ( \cos 30^\circ \times \cos 60^\circ \right ) \times \left ( \cos 40^\circ \times \cos 50^\circ \right )$

$= \left ( \cos 10^\circ \times \sin 10^\circ \right ) \times \left ( \cos 20^\circ \times \sin 20^\circ \right ) \times \left ( \cos 30^\circ \times \sin 30^\circ \right ) \times \left ( \cos 40^\circ \times \sin 40^\circ \right )$

$= \left ( \frac {1}{2} \right )^4 \times \sin 20^\circ \times \sin 40^\circ \times \sin 60^\circ \times \sin 80^\circ$

$= \frac {1}{16} \times \sin 60^\circ \times \frac {1}{4} \times \left (4 \sin 20^\circ \times \sin 40^\circ \times \sin 80^\circ \right )$

$= \frac {1}{64} \times \sin 60^\circ \times \left (4 \sin 20^\circ \times \sin (60-20)^\circ \times \sin (60+20)^\circ \right )$

$= \frac {1}{64} \times \sin 60^\circ \times \sin 60^\circ$

$= \frac {1}{64} \times \left ( \frac {\sqrt 3}{2} \right )^2$

$= \large \boxed{\frac {3}{256}}$

First of all we seperate the known quantities :-

$(cos30\quad \times \quad cos60)$

Then by using the formula :-

$4\quad \times \quad [CosA\quad \times \quad Cos(60-A)\quad \times \quad Cos(60+A)]\quad =\quad Cos3A$

We do the grouping as follows --- >

$Cos10\quad \times \quad Cos50\quad \times \quad Cos70\quad \quad ------\quad (Group\quad 1)\\ Cos20\quad \times \quad Cos40\quad \times \quad Cos80\quad \quad ------\quad (Group\quad 2)$

Now for group 1...

$Cos10\quad \times \quad Cos50\quad \times \quad Cos70\quad \times \quad \frac { 4 }{ 4 } \quad ------(1)\\ As\quad we\quad know\quad from\quad the\quad formula\quad stated\quad above\quad (1)\quad =\quad \frac { Cos(3\times 10) }{ 4 } \quad =\quad \frac { Cos30 }{ 4 }$

Similarly for group 2...

$Cos20\quad \times \quad Cos40\quad \times \quad Cos80\quad \times \quad \frac { 4 }{ 4 } \quad \quad ------(2)\\ =\quad \frac { Cos(20\times 3) }{ 4 } \quad =\quad \frac { Cos60 }{ 4 }$

Now the simplest part :D multiply and the results we have...

$Cos30\quad \times \quad Cos60\quad \times \quad \frac { Cos30 }{ 4 } \quad \times \quad \frac { Cos60 }{ 4 } \quad =\quad \boxed { \frac { 3 }{ 256 } }$

Phew.. That was a lot of work done on LaTeX.This is my second time on LaTeX so please excuse me for any mistakes if there are any.Hope this helps.