Question

Prove: Cosec^2 pi/5 + Cosec^2 2pi/5 = 4​

Answer 1

To prove : $cosec^2\frac{\pi}{5}+cosec^2\frac{2\pi}{5}=4$

solution : we know,

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

so cosec(π/5) = $\frac{4}{\sqrt{10-2\sqrt{5}}}$

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

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

now LHS = cosec²(π/5) + cosec²(2π/5)

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

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

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

=$\frac{10+2\sqrt{5}}{5}+\frac{10-2\sqrt{5}}{5}$

= $\frac{10+2\sqrt{5}+10-2\sqrt{5}}{5}$

= 4 = RHS

hence cosec²(π/5) + cosec²(2π/5) = 4.

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