(sin75° + sin15°) / (sin75° - sin15°) =?

আমরা জানি, \( \sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2} \) এবং \( \sin C - \sin D = 2 \cos \frac{C+D}{2} \sin \frac{C-D}{2} \)।

সুতরাং,

\( \frac{\sin 75^\circ + \sin 15^\circ}{\sin 75^\circ - \sin 15^\circ} = \frac{2 \sin \frac{75^\circ + 15^\circ}{2} \cos \frac{75^\circ - 15^\circ}{2}}{2 \cos \frac{75^\circ + 15^\circ}{2} \sin \frac{75^\circ - 15^\circ}{2}} \)

\( = \frac{\sin \frac{90^\circ}{2} \cos \frac{60^\circ}{2}}{\cos \frac{90^\circ}{2} \sin \frac{60^\circ}{2}} \)

\( = \frac{\sin 45^\circ \cos 30^\circ}{\cos 45^\circ \sin 30^\circ} \)

\( = \frac{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}}{\frac{1}{\sqrt{2}} \cdot \frac{1}{2}} \)

\( = \frac{\frac{\sqrt{3}}{2\sqrt{2}}}{\frac{1}{2\sqrt{2}}} \)

\( = \frac{\sqrt{3}}{2\sqrt{2}} \cdot \frac{2\sqrt{2}}{1} \)

\( = \sqrt{3} \) 🥳

অতএব, \( \frac{\sin 75^\circ + \sin 15^\circ}{\sin 75^\circ - \sin 15^\circ} = \sqrt{3} \). 🤩