Sum and Difference Formulas

\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b

\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b

\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}

\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}

Use \sin(105º) = \sin(60º + 45º) = \sin 60º \cos 45º + \cos 60º \sin 45º = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2} + \sqrt{6}}{4}.

\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} = \frac{\frac{3}{4} - (-\frac{1}{2})}{1 + \frac{3}{4} \cdot (-\frac{1}{2})} = 2.

Use \sin(\frac{7\pi}{12}) = \sin(\frac{3\pi}{4} + \frac{\pi}{3}) = \sin(\frac{3\pi}{4}) \cos(\frac{\pi}{3}) + \cos(\frac{3\pi}{4}) \sin(\frac{\pi}{3}) = \frac{\sqrt{6} + \sqrt{2}}{4}.

Use the formula \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}. First, find \tan \alpha and \tan \beta using the definitions of sine and cosine.

\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta.

Use \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta. First, find \cos \alpha using \sin^2 \alpha + \cos^2 \alpha = 1.