Question

find the value of Sin10Sin30Sin70Sin50​

Answer 1

   

$\Rightarrow\ \sin10 \cdot \sin30 \cdot \sin50 \cdot \sin70 \\ \\ \Rightarrow\ \frac{1}{2} \cdot \sin10 \cdot \sin50 \cdot \sin70\ \ \ \ \ \ \ \ \ \ \ \ [\sin30=\frac{1}{2}] \\ \\ \Rightarrow\ \frac{\sin10 \cdot \sin50 \cdot \sin70}{2} \\ \\ \Rightarrow\ \frac{2\cos10 \cdot \sin10 \cdot \sin50 \cdot \sin70}{2\cos10 \times 2} \\ \\ \Rightarrow\ \frac{\sin20\cdot \sin50 \cdot \sin70}{4\cos10}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2\sin\theta \cdot \cos\theta=\sin(2\theta)]$

$\Rightarrow\ \frac{\sin20\cdot \sin50 \cdot \sin(90-20)}{4\cos10} \\ \\ \Rightarrow\ \frac{\sin20 \cdot \sin 50 \cdot \cos20}{4\cos10} \\ \\ \Rightarrow\ \frac{2\sin20 \cdot \cos20\cdot \sin50}{2 \times 4\cos 10} \\ \\ \Rightarrow\ \frac{\sin40\cdot \sin50}{8\cos 10} \\ \\ \Rightarrow\ \frac{\sin40\cdot \sin(90-40)}{8\cos 10} \\ \\ \Rightarrow\ \frac{2\sin40 \cdot \cos40}{2 \times 8\cos10} \\ \\ \Rightarrow\ \frac{\sin80}{16\cos10}$

$\Rightarrow\ \frac{\sin80}{16\cos(90-80)} \\ \\ \Rightarrow\ \frac{\sin80}{16\sin80} \\ \\ \Rightarrow\ \bold{\frac{1}{16}} \\ \\ \\ \\ \therefore\ \sin10 \cdot \sin30 \cdot \sin50 \cdot \sin70=\bold{\frac{1}{16}}$

$Thank\ you... \\ \\ \\ \\ \\ \mathfrak{\#adithyasajeevan}$