The value of- $2 \cos{\left ( \frac{π}{13} \right )} \cos{\left ( \frac{9 π}{13} \right )} + \cos{\left ( \frac{3 π}{13} \right )} + \cos{\left ( \frac{5 π}{13} \right )}$ is-

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ব্যাখ্যা: আমরা জানি, $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$।
সুতরাং, $2 \cos\left(\frac{9\pi}{13}\right) \cos\left(\frac{\pi}{13}\right) = \cos\left(\frac{10\pi}{13}\right) + \cos\left(\frac{8\pi}{13}\right)$।
এখন,
$\cos\left(\frac{10\pi}{13}\right) = \cos\left(\pi - \frac{3\pi}{13}\right) = -\cos\left(\frac{3\pi}{13}\right)$
$\cos\left(\frac{8\pi}{13}\right) = \cos\left(\pi - \frac{5\pi}{13}\right) = -\cos\left(\frac{5\pi}{13}\right)$
অতএব,
$2 \cos\left(\frac{\pi}{13}\right) \cos\left(\frac{9\pi}{13}\right) + \cos\left(\frac{3\pi}{13}\right) + \cos\left(\frac{5\pi}{13}\right) = -\cos\left(\frac{3\pi}{13}\right) - \cos\left(\frac{5\pi}{13}\right) + \cos\left(\frac{3\pi}{13}\right) + \cos\left(\frac{5\pi}{13}\right) = 0$।

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