Question

$If\cos {10}^o\cos {30}^o\cos {50}^o\cos {70}^o=x$. $Find$ $16x$

Answer 1

$\cos {10}^{\circ }\cos {30}^{\circ }\cos {50}^{\circ }\cos {70}^{\circ }=x$

$\cos {10}^{\circ }\times \frac{\sqrt{3}}{2}\times \cos {50}^{\circ }\cos {70}^{\circ }=x$

$\frac{\sqrt{3}}{2}\left(\frac{2}{2}\cos {10}^{\circ }\cos {50}^{\circ }\cos {70}^{\circ }\right)=x$

$\sqrt{3}\left(\cos {10}^{\circ }\left(\cos {120}^{\circ }+\cos {20}^{\circ }\right)\right)=4x$

$\sqrt{3}\left(\cos {10}^{\circ }\left(-\frac{1}{2}+\cos {20}^{\circ }\right)\right)=4x$

$\sqrt{3}\left(\cos {10}^{\circ }\left(2\cos {20}^{\circ }-1\right)\right)=8x$

$\sqrt{3}\left(2\cos {10}^{\circ }\cos {20}^{\circ }-\cos {10}^{\circ }\right)=8x$

$\sqrt{3}\left(\cos {30}^{\circ }+\cos {10}^{\circ }-\cos {10}^{\circ }\right)=8x$

$\frac{3}{2}=8x$

$16x=3$

Therefore, the value of $16x$ is 3.