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Question Number 143726 by mathdanisur last updated on 17/Jun/21

Commented by mr W last updated on 17/Jun/21

$$=\frac{3}{256}?$$

Commented by mathdanisur last updated on 17/Jun/21

$$yesdearSirsolutionpossibleplease..$$

Answered by mr W last updated on 17/Jun/21

$$\sin 10°\sin 80°=\frac{2\sin 10°\cos 10°}{2}=\frac{\sin 20°}{2}$$$$\sin 20°\sin 70°=\frac{2\sin 20°\cos 20°}{2}=\frac{\sin 40°}{2}$$$$\sin 30°\sin 60°=\frac{\sqrt{3}}{4}$$$$\sin 40°\sin 50°=\frac{2\sin 40°\cos 40°}{2}=\frac{\cos 10°}{2}$$$$P=\frac{\cos 10°\sin 20°\sin 40°\sqrt{3}}{2^5}$$$$=\frac{\cos 10°\sin (30°-10°)\sin (30°+10°)\sqrt{3}}{2^5}$$$$=\frac{\cos 10°({\cos }^{2}10°-3{\sin }^{2}10°)\sqrt{3}}{2^7}$$$$=\frac{\cos 10°(4{\cos }^{2}10°-3)\sqrt{3}}{2^7}$$$$=\frac{\cos 3\times 10°\sqrt{3}}{2^7}$$$$=\frac{\cos 30°\sqrt{3}}{2^7}$$$$=\frac{\sqrt{3}\sqrt{3}}{2^8}$$$$=\frac{3}{2^8}=\frac{3}{256}$$

Commented by mathdanisur last updated on 17/Jun/21

$$perfectdearSir,thankyousomuch..$$

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