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# Proof of the cosine angle addition identity

Sal proves the identity cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y). Created by Sal Khan.

## Want to join the conversation?

• Is there a real proof of the angle addition formula involving tangent?
• Once you prove the cosine and sine formulas, you can divide by them and simplify to get the tangent addition formula, as tan(x) is sin(x)/cos(x), so tan (x+y) = sin(x+y) / cos(x+y)
• What if the length of AD is not 1? Then how to prove it?
• If you make the length of AD an arbitrary length, say r, you will just have a bunch of rs floating around until they cancel out and you get the identity we want. This is similar to the reason that we usually use the unit circle to find the trig functions of any angle and not just a generic circle with radius r.
• Something's been bugging me and I know i must be thinking about this the wrong way. Wouldn't the Cos(x+y) always equal 0 in a right triangle? Cause x+y have to equal 90 and the cos of 90 is 0.
• You're right.
But we still use the addition formulas to solve other trigonometric equations where the two angles, x and y, wouldn't add to 90.
• Is it worth remembering this proofs?
• Remembering proofs is not important.

But, understanding the steps of the proof is important.
• why there is no tangent version of this?
• you can find tangent version:
tan(a+b) = sin (a+b) / cos(a+b)
sin (a+b) = sin(a).cos(b) + sin(b).cos(a)
cos(a+b) = cos(a).cos(b) - sin(a).sin(b)
tan(a+b) = (sin(a).cos(b) + sin(b).cos(a)) / (cos(a).cos(b) - sin(a).sin(b))
now divide numerator and denominator by same factor (cos(a).cos(b))
then
tan(a+b) = ( tan(a) + tan(b) ) / ( 1 - tan(a).tan(b) )
• If a statement says x is an acute angle, cosx=1/2. Is that enough to determine wether or not sinx is positive or negative?
• According to Princeton University, an acute angle is an angle whose measure is between 0° and 90° (or 0 and π/2 radians). This means that even without knowing that `cos(x) = 1/2` we already know that sin(x) is positive since sine of anything in that range will output a value between 0 and 1. Since we also know that `cos(x) = 1/2`, we can find that `sin(x) = √̅3/2`.

http://wordnetweb.princeton.edu/perl/webwn?s=acute%20angle
• How does this proof account for values for angle x and y when x or y is greater than or equal to 90? The issue I believe I am observing is that since in the proof, angle x is part of a right triangle, so x has to be less than 90 degrees, and the same can be said for angle y. So, in light of this observation, despite the elegance of this proof compared to others i have seen, isn't this proof flawed, in that it isn't inclusive of angles greater than or equal to 90, thus making it an invalid proof for all real number angle values of x and y?
• This way you, at best substitute one law for another, and that's completely different matter, at worst make unwarranted assumptions about validity of some angel-based arithmetic which you yet has to prove (circular reasoning). @weisbed1 made valid point. Proof supplied in this video proves that this formula works for x,y and x+y acute. Nothing more. I posted outline of simple, elementary general prove of cosine addition formula in tips and thanks.
• Why is there ^ in the transcript
• It's a way to write 'to the power of'
For example: a^2 = a²
(1 vote)
• The sin addition identity makes sense because we solve for the entire y axis. However, cos solves for only part of the x axis? Why are we solving for segment AF and not for AB?
• On the sine angle identity, you never solved for the entire y axis. Think back to the unit circle, if you have learned it. The angle is x + y, which is equal to A. Remember what sine is. Sine isn't the entire y-axis. Instead, it is the opposite side divided by the hypotenuse. Triangle DAF has angle A, along with a hypotenuse of 1. Since the hypotenuse is 1, you are just dividing the opposite side by 1 to get the sine, which means the the sine is the length of the opposite side in this case. That is why DF is the sine of x + y.

As for this problem, you now have to remember what cosine is. Cosine is the adjacent side divided by the hypotenuse. Again, the hypotenuse of the triangle with x + y is 1. Therefore, you are just dividing the side adjacent to the angle by 1, so the cosine is equal to the adjacent side, which is AF.
(1 vote)
• Hi, can you make a video or show how to prove of the tangent identity?
• Let's assume we've already established the addition identities for sine and cosine:

sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
cos(A+B) = cos(A)cos(B) - sin(A)sin(B).

Using the fact that tangent equals sine over cosine, and dividing every term on the top and bottom by cos(A)cos(B) gives us:

tan(A+B) = sin(A+B)/cos(A+B)
= [sin(A)cos(B) + cos(A)sin(B)]/[cos(A)cos(B) - sin(A)sin(B)]
= [sin(A)/cos(A) + sin(B)/cos(B)]/[1 - sin(A)/cos(A) * sin(B)/cos(B)]
= [tan(A) + tan(B)]/[1 - tan(A)tan(B)].