Question

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

• A. $\frac{1}{4}$
• B. $\frac{1}{16}$
• C. $\frac{3}{4}$
• D. $\frac{5}{16}$

Answer 1

The correct option is D $\frac{5}{16}$

$\sin {36}^o\sin {72}^o\sin {108}^o\sin {144}^o$

$=\sin {36}^o\sin {72}^o\sin ({180}^o-{72}^o)\sin ({180}^o-{36}^o)$

$=\sin {36}^o\sin {72}^o\sin \left[\right({90}^o\times 2)-{72}^o]\left[\sin \right({90}^o\times 2)-{36}^o]$

$=\sin {36}^o\sin {72}^o\sin {72}^o\sin {36}^o$

$={\sin }^2{36}^o{\sin }^2{72}^o$

$=[\frac{\sqrt{(10-2\sqrt{5})}}{4}{]}^2\times [\frac{\sqrt{(10+2\sqrt{5})}}{4}{]}^2$

$=\frac{(10-2\surd 5)}{16}\times \frac{(10+2\surd 5)}{16}$

$=\frac{(100-20)}{(16\times 16)}$

$=\frac{80}{16\times 16}$

$=\frac{5}{16}$

Therefore, correct answer is $\frac{5}{16}$