# Sum and Difference Identities

## The Bhaskaracharya sum and difference formulas

### Sum Identities

$\mathrm{sin}\left(u+v\right)=\mathrm{sin}\left(u\right)\mathrm{cos}\left(v\right)+\mathrm{cos}\left(u\right)\mathrm{sin}\left(v\right)$

$\mathrm{cos}\left(u+v\right)=\mathrm{cos}\left(u\right)\mathrm{cos}\left(v\right)-\mathrm{sin}\left(u\right)\mathrm{sin}\left(v\right)$

### Difference Identities

$\mathrm{sin}\left(u-v\right)=\mathrm{sin}\left(u\right)\mathrm{cos}\left(v\right)-\mathrm{cos}\left(u\right)\mathrm{sin}\left(v\right)$

$\mathrm{cos}\left(u-v\right)=\mathrm{cos}\left(u\right)\mathrm{cos}\left(v\right)+\mathrm{sin}\left(u\right)\mathrm{sin}\left(v\right)$

Example: 1

Rewrite in a simpler form using a trigonometric identity.

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

Using the difference formula for the sine, we get

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

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

Example: 2

Evaluate: $\mathrm{tan}\left(15°\right)$

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

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

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

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

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

$=2-\sqrt{3}$