Question

The value of $\sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)$ is

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

Answer 1

The correct option is D

$\frac{5}{16}$

Explanation for the correct option

The given trigonometric expression: $\sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)$
$$\begin{gathered} \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)=\sin \left(36°\right)\sin \left(72°\right)\sin \left(180°-72°\right)\sin \left(180°-36°\right) \\ \Rightarrow \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)=\sin \left(36°\right)\sin \left(72°\right)\sin \left(72°\right)\sin \left(36°\right)\left[\because \sin \left(180°-\theta \right)=\sin \theta \right] \\ \Rightarrow \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)={\sin }^2\left(36°\right){\sin }^2\left(72°\right) \\ \Rightarrow \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)={\left[\frac{1}{4}\sqrt{10-2\sqrt{5}}\right]}^2{\left[\frac{1}{4}\sqrt{10+2\sqrt{5}}\right]}^2 \\ \Rightarrow \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)=\left[\frac{10-2\sqrt{5}}{16}\right]\left[\frac{10+2\sqrt{5}}{16}\right] \\ \Rightarrow \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)=\left[\frac{{\left(10\right)}^2-{\left(2\sqrt{5}\right)}^2}{256}\right] \\ \Rightarrow \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)=\left[\frac{100-20}{256}\right] \\ \Rightarrow \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)=\left[\frac{80}{256}\right] \\ \Rightarrow \sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)=\frac{5}{16} \end{gathered}$$

Therefore, the value of $\sin \left(36°\right)\sin \left(72°\right)\sin \left(108°\right)\sin \left(144°\right)$ is $\frac{5}{16}$.

Hence, the correct option is (D).