Question

Sin2 pi/18+sin2 pi/9+sin2 7pi/18+sin2 4pi/9=2

Answer 1

$\underline{\textbf{To prove:}}$

$\mathsf{sin^2\dfrac{\pi}{18}+sin^2\dfrac{\pi}{9}+sin^2\dfrac{7\pi}{18}+sin^2\dfrac{4\pi}{9}=2}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Identity used:}}$

$\mathsf{1.\;sin\theta=cos\left(\dfrac{\pi}{2}-\theta\right)}$

$\mathsf{2.\;sin^2\dfrac{\pi}{2}+cos^2\dfrac{\pi}{2}=1}$

$\mathsf{Consider,}$

$\mathsf{sin^2\dfrac{\pi}{18}+sin^2\dfrac{\pi}{9}+sin^2\dfrac{7\pi}{18}+sin^2\dfrac{4\pi}{9}}$

$\textsf{Using identity (1), we get}$

$\mathsf{=cos^2\left(\dfrac{\pi}{2}-\dfrac{\pi}{18}\right)+cos^2\left(\dfrac{\pi}{2}-\dfrac{\pi}{9}\right)+sin^2\dfrac{7\pi}{18}+sin^2\dfrac{4\pi}{9}}$

$\mathsf{=cos^2\left(\dfrac{9\pi-\pi}{18}\right)+cos^2\left(\dfrac{9\pi-2\pi}{9}\right)+sin^2\dfrac{7\pi}{18}+sin^2\dfrac{4\pi}{9}}$

$\mathsf{=cos^2\dfrac{8\pi}{18}+cos^2\dfrac{7\pi}{18}+sin^2\dfrac{7\pi}{18}+sin^2\dfrac{4\pi}{9}}$

$\mathsf{=cos^2\dfrac{4\pi}{9}+cos^2\dfrac{7\pi}{18}+sin^2\dfrac{7\pi}{18}+sin^2\dfrac{4\pi}{9}}$

$\textsf{Rearranging terms, we get}$

$\mathsf{=\left(cos^2\dfrac{4\pi}{9}+sin^2\dfrac{4\pi}{9}\right)+\left(cos^2\dfrac{7\pi}{18}+sin^2\dfrac{7\pi}{18}\right)}$

$\textsf{Using identity (2), we get}$

$\mathsf{=1+1}$

$\mathsf{=2}$

$\implies\boxed{\mathsf{sin^2\dfrac{\pi}{18}+sin^2\dfrac{\pi}{9}+sin^2\dfrac{7\pi}{18}+sin^2\dfrac{4\pi}{9}=2}}$