# Prove sin 75 = [root(6)+root(2)]/4

\begin{aligned} \sin 75^{\circ} &=\sin \left(30^{\circ}+45^{\circ}\right) \\ \sin (a+b) &=\sin a \cos b+\cos a \sin b \\ \sin 75^{\circ} &=\sin 30^{\circ} \cos 45^{\circ}+\cos 30^{\circ} \sin 45^{\circ} \\ &=\frac{1}{2} \times \frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{2}} \\ &=\frac{1+ \sqrt{3}}{2\sqrt{2}} \\ &=\frac{\sqrt{2}}{\sqrt{2}} \times \frac{1+\sqrt{3}}{2 \sqrt{2}} \\ &=\frac{\sqrt{2}+\sqrt{6}}{4} \\ \sin 75^{\circ} &=\frac{\sqrt{6}+\sqrt{2}}{4} \\ \text { Hence proved. } \end{aligned}

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