More proofs? Couldn't we get a root canal, or listen to babies crying on loop for 10 hours instead?
We could do those, but they'd be a lot more painful than this will be. We did all the heavy lifting last time; finding cosine follows the same logic as sine, we just use different lines in our answer, which changes the final formula a little bit.
Once again, we start with:
As before, we'll play match-breaker to the numerator, breaking it up and shuffling the pieces around.
This time we have some subtraction, but it is the same idea. Now we can relate 
and 
to the angle α using trig functions.
Now we rearrange these functions for the lines, and stick them back into our formula equation. We've got déjà vu going on in here.
One more set of substitutions with β and we have the addition formula for cosine.
cos (α + β) = cos (α) cos (β) – sin (α) sin (β)
Think of sine and cosine as rival softball teams. Each angle sticks with their team—sine with sine, cosine with cosine—even if they have friends on the other team. And sine is having a very bad year; they've somehow scored negative points throughout the season. No one is even sure how they did it.
Sample Problem
Find the exact value of cos (-135°).
Don't panic at that negative angle. We've got cosine, and cosine is an even function f(-x) = f(x).
Our answer will be the same as cos 135°. See, no need to panic.
There are a few ways we can get to 135°. Using our special triangles, we can use 135° = 30° + 45° + 60°, which would involve using the addition formula twice. That doesn't sound like so much fun, but there is another option.
135° is also equal to 90° + 45°. Taking a quick peek at the graphs of y = sin x and y = cos x, or at the unit circle, shows that sin (90°) is 1 and cos (90°) is 0. That's easy enough to work with, so let's use 90° + 45°
cos (90 + 45) = cos (90) cos (45) – sin (90) sin (45)
This formula might be even easier to use than sine's.
In A Related Formula
We still need to find the addition formula for tangent. It gets easier from here, though, we promise. Usually tangent makes things harder for us, but maybe it's warming up to us? The trick of it is that tangent is always equal to sine over cosine.
Like a butterfly bursting forth from its cocoon, we'll substitute in the addition formulas for sine and cosine to get a beautiful new equation.
Okay, it's not a beautiful butterfly yet. More like a sleepy wasp. We can fix this, though. We can cancel some terms, and trade in some of our sin α's, by dividing everything by cos α.
That's looking a little better, but it could use some more help. Let's divide the top and bottom by cos β now:
Maybe not all formulas can be butterflies. But at least we can use this one to find more angles.
Sample Problem
Simplify 
.
Math Wars: The Radians Strike Back. And this time, it's brought a variable with it. We can still apply the formula, though.
Now we just take a special consultation call with the 30-60-90 triangle to simplify as much as we can. And unlike calling a psychic hotline, it doesn't cost us $4.99 a minute.
This may look more complicated, but it is actually simplified. We can't really do much with a trig function when it has addition stuck inside it.
Summary
We've got three formula-la-las for adding angles together.
- sin (α + β) = sin (α) cos (β) + cos (α) sin (β)
- cos (α + β) = cos (α) cos (β) – sin (α) sin (β)
If your brain is completely strapped for space, remembering the sine and cosine addition functions are the most important. We can find the tangent addition formula (and some others, shhh) using them.
But really, you must have some room left, or some spare clutter you can toss out. Do you really need to be able to quote the entirety of The Princess Bride forwards and backwards? (Actually, yes we do.)