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The value of $\sin 36^{\large\circ}\sin 72^{\large\circ}\sin 108^{\large\circ}\sin 144^{\large\circ}$ is equal to

$(a)\;\large\frac{1}{16}$$\qquad(b)\;\large\frac{1}{4}$$\qquad(c)\;\large\frac{3}{4}\qquad$$(d)\;\large\frac{5}{16}$

1 Answer

$\sin 36^{\large\circ}\sin 72^{\large\circ}\sin 108^{\large\circ}\sin 144^{\large\circ}$
$\Rightarrow \sin 36^{\large\circ}\sin 72^{\large\circ}.\sin 36^{\large\circ}.\sin 72^{\large\circ}$
$\Rightarrow \sin^2 36^{\large\circ}\sin ^272^{\large\circ}$
$\Rightarrow \large\frac{10-2\sqrt 5}{16}.\frac{10+2\sqrt 5}{16}$
$\Rightarrow \large\frac{(10-2\sqrt 5)(10+2\sqrt 5)}{16\times 16}$
$\Rightarrow \large\frac{(10)^2(2\sqrt 5)^2}{16\times 16}$
$\Rightarrow \large\frac{100-20}{16\times 16}$
$\Rightarrow \large\frac{80}{16\times 16}$
$\Rightarrow \large\frac{5}{ 16}$
Hence (d) is the correct option.
answered Oct 16, 2013 by sreemathi.v
 

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