Question 15.4.25: Prove that sin π/5 sin 2π/5 sin 3π/5 sin 4π/5=5/16....
Prove that \sin \frac{\pi}{5} \sin \frac{2 \pi}{5} \sin \frac{3 \pi}{5} \sin \frac{4 \pi}{5}=\frac{5}{16} .
Step-by-Step
We have
\begin{array}{l} \text { LHS }=\sin \frac{\pi}{5} \sin \frac{2 \pi}{5} \sin \frac{3 \pi}{5} \sin \frac{4 \pi}{5}\\ \\=\sin \frac{\pi}{5} \sin \frac{2 \pi}{5} \sin \left(\pi-\frac{2 \pi}{5}\right) \sin \left(\pi-\frac{\pi}{5}\right)\\ \\=\sin ^{2} \frac{\pi}{5} \sin ^{2} \frac{2 \pi}{5} \quad[\because \sin (\pi-\theta)=\sin \theta] \\ \\=\left(\sin 36^{\circ}\right)^{2} \times\left(\sin 72^{\circ}\right)^{2}\\ \\=\left(\sin 36^{\circ}\right)^{2} \times\left(\cos 18^{\circ}\right)^{2}\\ \\ \left[ \because \sin 72^{\circ} = \sin ( 90^{\circ} -18^{\circ}) = \cos 18^{\circ} \right] \\ \\=\frac{(10-2 \sqrt{5})}{16} \times \frac{(10+2 \sqrt{5})}{16}=\frac{(100-20)}{(16 \times 16)}\\ \\{\left[\because \sin 36^{\circ}=\frac{\sqrt{10-2 \sqrt{5}}}{4} \text { and } \cos 18^{\circ}=\frac{\sqrt{10+2 \sqrt{5}}}{4}\right] }\\ \\=\frac{80}{(16 \times 16)}=\frac{5}{16}=\text { RHS. }\end{array}Related Answered Questions
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