7.2: Addition and Subtraction Formulas
We know how to find $\sin(x)$: just find how far you rotate on the unit circle.
That $x$ value can be decomposed into a sum of two values $x = s + t$. Is there a way to find $\sin(x) = \sin(s + t)$ in terms of just $s$ and $t$ plugged in?
The answer is yes:
For sine:
\begin{align}
\sin(s + t) &= \sin s \cos t + \cos s \sin t\\[0.5em]
\sin(s - t) &= \sin s \cos t - \cos s \sin t
\end{align}
For cosine:
\begin{align}
\cos(s + t) &= \cos s \cos t - \sin s \sin t \\[0.5em]
\cos(s - t) &= \cos s \cos t + \sin s \sin t
\end{align}
For tangent:
\begin{align}
\tan(s + t) &= \frac{\tan s + \tan t}{1 - \tan s \tan t}\\[0.5em]
\tan(s - t) &= \frac{\tan s - \tan t}{1 + \tan s \tan t}
\end{align}
We will prove $\cos(s + t) = \cos s \cos t - \sin s\sin t$ in class.
Find \[\cos 70^\circ \qquad\qquad \cos \frac{\pi}{12} \qquad\qquad \sin20^\circ\cos40^\circ + \cos20^\circ \sin40^\circ\]
Prove \[\frac{1 + \tan x}{1 - \tan x} = \tan\left(\frac{\pi}{4} + x\right)\]
Prove \[\cos\left(\dfrac{\pi}{2} - u\right) = \sin u\]
Prove \[\cos(x + y)\cos(x - y) = \cos^2x - \sin^2y\]
Hint: apply Pythagorean identities.