\[ \cos 75^\circ = \cos(45^\circ + 30^\circ) \]

\[ \cos(45^\circ + 30^\circ) = \cos 45^\circ\,\cos 30^\circ \;-\; \sin 45^\circ\,\sin 30^\circ \]

\[ = \frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2} \;-\; \frac{\sqrt2}{2}\cdot\frac{1}{2} = \frac{\sqrt6-\sqrt2}{4}. \]

\[ \cos 15^\circ = \cos(45^\circ - 30^\circ) \]

\[ \cos(45^\circ - 30^\circ) = \cos 45^\circ\,\cos 30^\circ \;+\; \sin 45^\circ\,\sin 30^\circ \]

\[ = \frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2} \;+\; \frac{\sqrt2}{2}\cdot\frac{1}{2} = \frac{\sqrt6+\sqrt2}{4}. \]

\[ \tan 105^\circ = \tan(60^\circ + 45^\circ) \]

\[ \tan(60^\circ + 45^\circ) = \frac{\tan 60^\circ + \tan 45^\circ}{\,1 - \tan 60^\circ\,\tan 45^\circ\,} = \frac{\sqrt3 + 1}{\,1 - \sqrt3\,} \]

\[ = -\,\frac{\sqrt3 + 1}{\sqrt3 - 1} = -\,\frac{(\sqrt3 + 1)^2}{(\sqrt3 - 1)(\sqrt3 + 1)} = -\,\frac{3 + 2\sqrt3 + 1}{2} = -\,(2+\sqrt3). \]

\[ \sin 15^\circ = \sin(45^\circ - 30^\circ) \]

\[ = \sin 45^\circ\,\cos 30^\circ \;-\; \cos 45^\circ\,\sin 30^\circ = \frac{\sqrt6-\sqrt2}{4}. \]

\[ \frac{1+\sin 15^\circ}{1-\sin 15^\circ} = \frac{\,1+\frac{\sqrt6-\sqrt2}{4}\,}{\,1-\frac{\sqrt6-\sqrt2}{4}\,} = \frac{\,4+\sqrt6-\sqrt2\,}{\,4-\sqrt6+\sqrt2\,}. \]

\[ \sin\frac{5\pi}{12} = \sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right) \]

\[ = \sin\frac{\pi}{4}\cos\frac{\pi}{6} + \cos\frac{\pi}{4}\sin\frac{\pi}{6} \]

\[ = \frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2} + \frac{\sqrt2}{2}\cdot\frac12 = \frac{\sqrt2}{4}(\sqrt3+1). \]