 Alright everyone, welcome to Tutor Terrific, this is part 3 of our 3-part series on deriving the sum and difference formulas. Today we are going to specifically derive the formulas for tangent of the sum of two angles and the tangent of the difference between two angles. These formulas again with the other two parts of the series are amazing because they are the foundation for so much of analytic trigonometry. A bunch of other formulas can be derived from these guys. Now, here are the formulas we are going to work on today. We are going to start by deriving tangent of u plus v which is the tangent of u plus the tangent of v over 1 minus the product of tangent u and tangent v. Then we are going to derive based on this result tangent u minus v which is equal to tangent u minus tangent v over 1 plus the product of tangent u, tangent v. As you can see the signs have just flip flopped. So here is how we are going to begin. Knowing that tangent is defined as sine over cosine with the quotient identity, I am going to write tangent of u plus v as the sine of u plus v over the cosine of u plus v. This is the quotient identity and this is completely valid. Next, I am going to take these two terms and I am going to expand them using the sum and difference formulas I have derived in part one and part two of this three part series. So this is equal to sine u cosine v plus cosine u sine v. That is what is going on on top. In part one, I have cosine u cosine v minus sine u sine v. These are those two formulas written out. Now the only unintuitive part in this process is what I am going to do right now. I am going to divide both the numerator and denominator by the following product. Cosine u cosine v. So both the numerator and denominator will themselves become fractions with cosine u cosine v as their denominators. So what does that look like? Well, it looks ugly but we are going to write it anyway. Sine u cosine v plus cosine u sine v all over cosine u cosine v. Now the denominator is cosine u cosine v minus sine u sine v all over cosine u cosine v. Oh my gosh, this is getting out of hand it seems. But not to worry, it is going to start simplifying right now. Both the top and the bottom can be written as two separate fractions. You are going to see stuff start to cancel when we do this. So let's begin. So I am going to write this first fraction as sine u cosine v over cosine u cosine v. I am going to add to that the other fraction cosine u sine v over cosine u cosine v. You can see stuff that is going to cancel. Next on the denominator we have cosine u cosine v over the same thing cosine u cosine v minus, in the last part we make that its own fraction, sine u sine v over cosine u cosine v. Sorry guys, my gimbal seemed to have a freak out moment there. It got too hot. So back to where we were we are going to cancel some things in each of these four mini fractions. So in the first fraction we see that the cosine v's cancel. In the second fraction we see the cosine u's cancel. Down below everything cancels because we have cosine u cosine v over cosine u cosine v. And in the last fraction nothing cancels. So now I am going to erase all this and write what we have left after these cancellations. Alright, here is what we have now. I took out everything that cancels and you can see that the third fraction, the first one on bottom, everything cancels so you have one left. Okay, so now again remind yourself of where we started this whole derivation at tangent u plus v. What we have at the top, the sine of u over cosine u, well using the quotient rule for tangents, quotient identity rather, well that's just tangent u. Okay, I can start to see this shaping up. Sine v over cosine v that will be tangent v. On bottom we will have the one minus. Now let's talk about this. If we split this fraction up into two fractions multiplied by each other, you can see that the first fraction is sine u over cosine u, that's tangent u just like on top. And same with the second fraction, sine v over cosine v, that's tangent v. So you have one minus tangent u tangent v. Alright, we have derived the formula for tangent u plus v. Now the trick that we are going to do to get tangent u minus v is the same trick we did for part two for the sine u minus v formula. We will rewrite tangent u minus v, which is what we want to derive as tangent u plus negative v. These are equivalent, but now we have the plus so we can reuse our result from the previous step, the previous formula duration, but we will plug in a negative v instead of a positive v wherever a v exists. So tangent u plus negative v is equal to tangent u, tangent u plus tangent negative v over one minus tangent u tangent negative v. Okay, as before, the second part and the first part, we need to discuss the odd and even nature of all the trick functions that have a negative angle to evaluate. It's just tangent negative v both on top and bottom. Now as it turns out, tangent is odd. And so if I plug in negative v, that is equal to negative tangent of positive v. So I'm going to rewrite tangent u so tangent negative v is equal to negative tangent v. And on bottom we have one minus tangent u negative tangent v. Okay, so on top we're set, but on bottom I'm going to move that negative out in front so I don't get confused here and think I'm subtracting when I'm actually multiplying. These two negatives will combine to make a positive. So all in all we'll have that the tangent of u minus v is equal to the tangent of u minus the tangent of v over one plus tangent u tangent v. And we have derived that formula as well. So we have both sum and difference formulas for tangent and we are all finished. Thank you for watching all three parts of this sum and difference formulas three part series. And this is Falconator signing out.