Question

find the value of

cos 10⁰ cos 50⁰ cos 30⁰ cos 70⁰

Answer 1

$$\begin{gathered} \cos 30.(\cos 50.\cos 10).\cos 70 \\ =\frac{\sqrt{3}}{2}.(\cos 50.\cos 10).\cos 70 \\ As2\cos x\cos y=\cos (x+y)+\cos (x-y) \\ Hence\frac{\sqrt{3}}{2}.(\cos 50.\cos 10).\cos 70=\frac{\sqrt{3}}{4}\left(2\cos 50\cos 10\right).\cos 70 \\ =\frac{\sqrt{3}}{4}(\cos 60+\cos 40).\cos 70 \\ =\frac{\sqrt{3}}{8}\cos 70+\frac{\sqrt{3}}{8}\left(2\cos 40\cos 70\right) \\ =\frac{\sqrt{3}}{8}\cos 70+\frac{\sqrt{3}}{8}(\cos 110+\cos 30) \\ =\frac{\sqrt{3}}{8}\cos 70+\frac{\sqrt{3}}{8}\cos (180-70)+\frac{\sqrt{3}}{8}\cos 30 \\ =\frac{\sqrt{3}}{8}\cos 70-\frac{\sqrt{3}}{8}\cos 70+\frac{\sqrt{3}}{8}\cos 30 \\ =\frac{\sqrt{3}}{8}\times \frac{\sqrt{3}}{2}=\frac{3}{16} \end{gathered}$$