Question

prove that : sin pi/5 * sin 2pi/5 * sin 3pi/5 * sin 4pi/5 = 5/16

Answer 1

We have to prove that sin(π/5) sin(2π/5) sin(3π/5) sin(4π/5) = 5/16

Proof : LHS = sin(π/5) sin(2π/5) sin(3π/5) sin(4π/5)

= sin(π/5) sin(2π/5) sin(π - 2π/5) sin(π - π/5)

= sin(π/5) sin(2π/5) sin(2π/5) sin(π/5)

= sin²(π/5) sin²(2π/5)

we know, sin(π/5) = $\frac{\sqrt{10-2\sqrt{5}}}{4}$

and

sin(2π/5) = $\frac{\sqrt{10+2\sqrt{5}}}{4}$

so sin²(π/5) sin²(2π/5)

= $\left(\frac{\sqrt{10-2\sqrt{5}}}{4}\right)^2\times\left(\frac{\sqrt{10+2\sqrt{5}}}{4}\right)^2$

= $\frac{(10-2\sqrt{5})(10+2\sqrt{5})}{16\times16}$

= $\frac{(10^2-(2\sqrt{5})^2}{16\times16}$

= $\frac{100-20}{16\times16}$

= $\frac{80}{16\times16}$

= $\frac{5}{16}$ = RHS

hence proved.