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Current time:0:00Total duration:6:55

Proof of the tangent angle sum and difference identities

CCSS.Math:

Video transcript

in this video i'm going to assume that you already know a few things and we've covered this we've proved this in other videos that sine of x plus y is equal to sine of x cosine y plus and then you swap the cosines and the sines cosine of x sine y and that cosine of x plus y is equal to cosine x cosine y minus sine x sine y once again we've proven this in these in other videos and then there's some other properties we know of cosine and sine that we have looked at in other videos cosine of negative x is equal to cosine of x and that sine of negative x is equal to negative sine of x and that of course the tangent of something is defined as sine over cosine of that something now with that out of the way i want to come up with a formula for tangent of x plus y expressed just in terms of tangent of x and tangent of y you can view it as the analog for what we did up here for sine and cosine well the immediate thing that you might recognize is that tangent of x plus y based on the definition of tangent is the same thing as sine of x plus y over cosine of x plus y and what's that going to be equal to well we know that sine of x plus y can be expressed this way so let me write that down so that's going to be sine of x cosine y plus cosine of x sine of y and then and actually so that we can save a little bit of writing i'm going to awkwardly write make the line here down here because we're going to we're going to put something here in a second but i think you'll get the idea so that's going to be that over cosine of x plus y which is this expression when you just express it in terms of cosines of x and cosines of y and sines of x and sines of y so let me write it here so you're going to have cosine of x cosine y minus sine of x and then sine y now we want to express everything in terms of tangents of x's and y's and so it might make sense here to say all right well we know tangent is sine over cosine so what if we were to divide both the numerator and the denominator by some expression that can start to make the numerator and denominator express in terms of tangents and i will cut a little bit to the chase here so in the numerator what i can do is and i'm going to do this just in the numerator and then i'm going to do it in the denominator as well i'm going to divide the numerator by cosine of x cosine y and of course i can't just divide the numerator by cosine of x cosine y that would change the value of the this rational expression i have to do that to the denominator as well so i know this is a very complex looking fraction here but it's going to simplify in a second so i'm also going to divide the denominator by cosine of x cosine of y and now let's see if we can simplify this in certain ways on the in the numerator we can see that this cosine y cancels with this cosine y and so that first term becomes write in another color here so this sine of x over cosine of x and so the numerator i can say this is going to be equal to sine of x over cosine of x is tangent of x and then this second term here we can see that this cosine of x cancels with this cosine of x and so we're left with sine of y over cosine of y which is of course tangent of y so plus tangent of y and then all of that is going to be over now we can look at the denominator so this first term here we can see the cosine of x cancels with the cosine of x and the cosine of y cancels out with the cosine of y so you could view this first term here when you divide by this cosine of x cosine y it just becomes one and then we're going to have the minus and now this second term is interesting we have sine of x over cosine of x sine of y over cosine of y so sine of x over cosine of x that over there is tangent of x and then sine of y over cosine of y that's tangent of y so this is going to be tangent of x times tangent of y and just like that we have come up with an expression for tangent of x plus y that just deals with tangent of x's and tangent of ys now the next question you might say well all right that's great for tangent of x plus y but what about tangent of x minus y well here we just have to recognize a little bit of what we've seen before let me write it over here tangent of negative x is equal to sine of negative x over cosine of negative x and what's that going to be equal to and i know i'm running out of space this is going to be equal to sine of negative x is the same thing as negative sine of x negative sine of x and then cosine of negative x is just cosine of x well this is just the negative of the tangent of x so this is negative tangent of x and the reason why that is useful is i can rewrite this as being let me write it here this is the same thing as tangent of x plus negative y so everywhere we saw a y here we can replace it with a negative y so this is going to be equal to tangent of x plus the tangent of negative y all of that over 1 minus the tangent of x times the tangent of negative y well we know the tangent of negative y is the same thing as the negative tangent of y and we know that over here as well so this could just be we could write the tangent of y here and then the negative would turn this into a plus and so just to write everything neatly we know that also the tangent of x minus y can be rewritten as tangent of x minus tangent of y all of that over 1 plus tangent of x and tangent of y