cos frac{pi}{7} - cos frac{2pi}{7} + cos frac | vedclass

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(D) माना $S = \cos \frac{\pi}{7} - \cos \frac{2\pi}{7} + \cos \frac{3\pi}{7} - \cos \frac{4\pi}{7} + \cos \frac{5\pi}{7} - \cos \frac{6\pi}{7}$ है।गुण

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(D) माना $S = \cos \frac{\pi}{7} - \cos \frac{2\pi}{7} + \cos \frac{3\pi}{7} - \cos \frac{4\pi}{7} + \cos \frac{5\pi}{7} - \cos \frac{6\pi}{7}$ है।गुणधर्म $\cos(\pi - 	heta) = -\cos 	heta$ का उपयोग करने पर,हमें प्राप्त होता है $\cos \frac{6\pi}{7} = -\cos \frac{\pi}{7}$,$\cos \frac{5\pi}{7} = -\cos \frac{2\pi}{7}$,और $\cos \frac{4\pi}{7} = -\cos \frac{3\pi}{7}$।इन मानों को व्यंजक में प्रतिस्थापित करने पर:$S = \cos \frac{\pi}{7} - \cos \frac{2\pi}{7} + \cos \frac{3\pi}{7} - (-\cos \frac{3\pi}{7}) + (-\cos \frac{2\pi}{7}) - (-\cos \frac{\pi}{7})$$S = \cos \frac{\pi}{7} - \cos \frac{2\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{3\pi}{7} - \cos \frac{2\pi}{7} + \cos \frac{\pi}{7}$$S = 2 \cos \frac{\pi}{7} - 2 \cos \frac{2\pi}{7} + 2 \cos \frac{3\pi}{7}$।$2 \sin \frac{\pi}{7}$ से गुणा और भाग करने पर:$S = \frac{2 \sin \frac{\pi}{7} \cos \frac{\pi}{7} - 2 \sin \frac{\pi}{7} \cos \frac{2\pi}{7} + 2 \sin \frac{\pi}{7} \cos \frac{3\pi}{7}}{\sin \frac{\pi}{7}}$$2 \sin A \cos B = \sin(A+B) + \sin(A-B)$ सूत्र का उपयोग करने पर:$S = \frac{\sin \frac{2\pi}{7} - (\sin \frac{3\pi}{7} - \sin \frac{\pi}{7}) + (\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7})}{\sin \frac{\pi}{7}}$$S = \frac{\sin \frac{2\pi}{7} - \sin \frac{3\pi}{7} + \sin \frac{\pi}{7} + \sin \frac{4\pi}{7} - \sin \frac{2\pi}{7}}{\sin \frac{\pi}{7}}$चूँकि $\sin \frac{4\pi}{7} = \sin(\pi - \frac{4\pi}{7}) = \sin \frac{3\pi}{7}$,इसलिए पद कट जाएंगे:$S = \frac{\sin \frac{\pi}{7}}{\sin \frac{\pi}{7}} = 1$।

$\cos \frac{\pi}{7} - \cos \frac{2\pi}{7} + \cos \frac{3\pi}{7} - \cos \frac{4\pi}{7} + \cos \frac{5\pi}{7} - \cos \frac{6\pi}{7} = $

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