If $$x= (\cos1°) (\cos2°) (\cos3°) .............(\cos89°)$$ and
$$y=(\cos 2°)(\cos 6°)(\cos 10°).............(\cos 86°)$$ Then what is the integer nearest to $$\left( \frac{2}{7} \right) \left( \log_2 \frac{y}{x} \right) ?$$
I tried putting it in $\Pi$ form but had no luck.
$$ x = \Bigl(\cos(1^{\circ})\cos(2^{\circ})\cdots\cos(44^{\circ})\Bigr) \cdot \cos(45^{\circ}) \cdot \Bigl(\sin(44^{\circ})\sin(43^{\circ})\cdots\sin(1^{\circ})\Bigr)\tag{1} $$ $$ =\dfrac{1}{\sqrt{2}} \cdot 2^{-44} \sin(2^{\circ})\sin(4^{\circ})\cdots\sin(88^{\circ}). $$ Next step: denote $$z=\sin(2^{\circ})\sin(4^{\circ})\cdots\sin(88^{\circ}).\tag{2}$$ Then $$ z = \Bigl(\sin(2^{\circ})\sin(4^{\circ})\cdots\sin(44^{\circ})\Bigr) \cdot \Bigl(\cos(44^{\circ})\cos(42^{\circ})\cdots\cos(2^{\circ})\Bigr)\tag{3} $$ $$ =2^{-22}\sin(4^{\circ})\sin(8^{\circ})\cdots\sin(88^{\circ}) $$ $$ =2^{-22}\cos(86^{\circ})\cos(82^{\circ})\cdots\cos(2^{\circ}) $$ $$ =2^{-22}y. $$ Therefore: $$ x = \dfrac{1}{\sqrt{2}}\cdot 2^{-44}\cdot 2^{-22}y = 2^{-66.5}y.\tag{4} $$ Hence $$ \log_2(y/x) = 66.5; $$ $$ \dfrac{2}{7}\log_2(y/x) = \dfrac{2}{7}\cdot \dfrac{133}{2} = 19.\tag{5} $$
