cos^2A+cos^2(A+pi/3)+cos^2(A-pi/3) এর মান কোনটি?

Explanation:

Another Explanation (5): সমাধান: 🤔 আমরা জানি, \(cos^2x = \frac{1 + cos2x}{2}\) তাহলে, \(cos^2A = \frac{1 + cos2A}{2}\) \(cos^2(A+\frac{\pi}{3}) = \frac{1 + cos(2A+\frac{2\pi}{3})}{2}\) \(cos^2(A-\frac{\pi}{3}) = \frac{1 + cos(2A-\frac{2\pi}{3})}{2}\) সুতরাং, \(cos^2A+cos^2(A+\frac{\pi}{3})+cos^2(A-\frac{\pi}{3})\) \(= \frac{1 + cos2A}{2} + \frac{1 + cos(2A+\frac{2\pi}{3})}{2} + \frac{1 + cos(2A-\frac{2\pi}{3})}{2}\) \(= \frac{3}{2} + \frac{1}{2}[cos2A + cos(2A+\frac{2\pi}{3}) + cos(2A-\frac{2\pi}{3})]\) আমরা জানি, \(cos(C+D) + cos(C-D) = 2cosCcosD\) তাহলে, \(cos(2A+\frac{2\pi}{3}) + cos(2A-\frac{2\pi}{3}) = 2cos2Acos(\frac{2\pi}{3})\) আবার, \(cos(\frac{2\pi}{3}) = -\frac{1}{2}\) সুতরাং, \(cos(2A+\frac{2\pi}{3}) + cos(2A-\frac{2\pi}{3}) = 2cos2A(-\frac{1}{2}) = -cos2A\) তাহলে, \(= \frac{3}{2} + \frac{1}{2}[cos2A - cos2A]\) \(= \frac{3}{2} + \frac{1}{2} \times 0\) \(= \frac{3}{2}\) ✨ অতএব, \(cos^2A+cos^2(A+\frac{\pi}{3})+cos^2(A-\frac{\pi}{3}) = \frac{3}{2}\) 🎉