Math, asked by Thanuj727, 1 year ago

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

Answers

Answered by MaheswariS
3

\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}}

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