Trigonometric Identities

A compound angle is any angle written as an algebraic sum or difference of two (or more) other angles, such as $\alpha+\beta$, $\alpha-\beta$, or $A+B+C$. It is tempting to guess that a trigonometric function of a compound angle simply "distributes" the way a linear function would — but $\sin,\cos,\tan$ are not linear: in general $f(x+y)\ne f(x)+f(y)$ and $f(kx)\ne kf(x)$ for a real number $k$. For instance $\cos(\alpha+\beta)$ is not the same as $\cos\alpha+\cos\beta$ — take $\alpha=\beta=90^\circ$: the left side is $\cos180^\circ=-1$, while the right side is $0+0=0$. Likewise $\sin(2\alpha)\ne2\sin\alpha$ and $\tan(3\alpha)\ne3\tan\alpha$ in general. So a trig ratio of a compound angle genuinely needs its own formula — we cannot shortcut it by applying the function to each piece separately. …