# 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?

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.