Question

$\sin 36°\sin 72°\sin 108°\sin 144°=$

• 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 answer:

Simplifying the given expression:

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

Therefore, the correct answer is option (D).