Sum and Difference Identities

Example: 1

Rewrite in a simpler form using a trigonometric identity.

$\sin \left(\frac{\pi }{6}\right)\cos \left(\frac{\pi }{4}\right)-\cos \left(\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}\right)$

Using the difference formula for the sine, we get

$\sin \left(\frac{\pi }{6}-\frac{\pi }{4}\right)=\sin \left(\frac{2\pi -3\pi }{12}\right)$

$=\sin \left(-\frac{\pi }{12}\right)$

Example: 2

Evaluate: $\tan \left(15°\right)$

$\tan \left(15°\right)=\tan \left(45°-30°\right)$

$=\frac{\tan \left(45°\right) - \tan \left(30°\right)}{1 + \tan \left(45°\right)\tan \left(30°\right)}$

$=\frac{1+\frac{\sqrt{3}}{3}}{1+1\cdot \frac{\sqrt{3}}{3}}$

$=\frac{3-\sqrt{3}}{3+\sqrt{3}}$

$=\frac{3-\sqrt{3}}{3+\sqrt{3}}\cdot \frac{3-\sqrt{3}}{3-\sqrt{3}}$

$=\frac{12-6\sqrt{3}}{6}$

$=2-\sqrt{3}$