1. The Six Angle Addition & Subtraction Formulas
The angle addition and subtraction formulas (also called sum and difference identities) express the sine, cosine, or tangent of a combined angle $\alpha \pm \beta$ in terms of the trigonometric functions of $\alpha$ and $\beta$ individually.
Cosine Formulas
$$ \cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta $$
$$ \cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta $$
Mnemonic: The sign in the cosine formula is opposite to the sign between the angles. Sum of angles → minus sign; difference of angles → plus sign.
Sine Formulas
$$ \sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta $$
$$ \sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta $$
Mnemonic: The sign in the sine formula matches the sign between the angles.
Tangent Formulas
$$ \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} $$
$$ \tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta} $$
Conditions of validity:
- $\tan\alpha$ must be defined: $\alpha \neq \frac{\pi}{2} + n\pi$ for any integer $n$.
- $\tan\beta$ must be defined: $\beta \neq \frac{\pi}{2} + n\pi$ for any integer $n$.
- The combined angle $\alpha + \beta$ (or $\alpha - \beta$) must not equal $\frac{\pi}{2} + k\pi$, which is equivalent to requiring the denominator $1 - \tan\alpha\tan\beta \neq 0$ (or $1 + \tan\alpha\tan\beta \neq 0$).
For example, if $\alpha = 60°$ and $\beta = 30°$, then $\alpha + \beta = 90°$ and $\tan 90°$ is undefined. Indeed, $1 - \tan 60°\tan 30° = 1 - \sqrt{3}\cdot\frac{1}{\sqrt{3}} = 0$.