(sin75^o +sin15^o)/(sin75^o-sin15^o) =?

আমরা জানি, \(sin(A) + sin(B) = 2sin(\frac{A+B}{2})cos(\frac{A-B}{2})\) এবং \(sin(A) - sin(B) = 2cos(\frac{A+B}{2})sin(\frac{A-B}{2})\)

অতএব, \(sin75^o + sin15^o = 2sin(\frac{75+15}{2})cos(\frac{75-15}{2}) = 2sin(45^o)cos(30^o)\)

এবং, \(sin75^o - sin15^o = 2cos(\frac{75+15}{2})sin(\frac{75-15}{2}) = 2cos(45^o)sin(30^o)\)

সুতরাং, \(\frac{sin75^o + sin15^o}{sin75^o - sin15^o} = \frac{2sin(45^o)cos(30^o)}{2cos(45^o)sin(30^o)} = \frac{sin(45^o)}{cos(45^o)} \cdot \frac{cos(30^o)}{sin(30^o)}\)

আমরা জানি, \(tan(45^o) = 1\), \(cos(30^o) = \frac{\sqrt{3}}{2}\) এবং \(sin(30^o) = \frac{1}{2}\)

তাহলে, \(\frac{sin75^o + sin15^o}{sin75^o - sin15^o} = 1 \cdot \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\) 🎉

অতএব, \(\frac{sin75^o + sin15^o}{sin75^o - sin15^o} = \sqrt{3}\) ✅