Find the value of $$8\cos10^\circ\cos20^\circ\cos40^\circ$$

Expression : $$8cos10^\circ cos20^\circ cos 40^\circ$$

Multiply $$sin(10^\circ)$$ in numerator and denominator.

= $$(2sin10^\circ cos10^\circ)\times(4 cos20^\circ cos 40^\circ)\times\frac{1}{sin10^\circ}$$

Similarly,

= $$(2sin10^\circ cos10^\circ)\times(2sin20^\circ cos20^\circ)\times(2sin40^\circ cos 40^\circ)\times\frac{1}{sin10^\circ sin20^\circ sin40^\circ}$$

Using, $$2sinA cosA=sin2A$$

= $$(sin20^\circ)\times(sin40^\circ)\times(sin80^\circ)\times\frac{1}{sin10^\circ sin20^\circ sin40^\circ}$$

= $$\frac{sin(80^\circ)}{sin(10^\circ)}$$

Also, $$sin(90^\circ-\theta)=cos\theta$$

= $$\frac{sin(90^\circ-10^\circ)}{sin(10^\circ)}=\frac{cos10^\circ}{sin10^\circ}$$

= $$cot10^\circ=tan80^\circ$$

=> Ans - (C)