### Video Transcript

Prove that sin 𝐴 minus two sin cubed 𝐴 over two cos cubed 𝐴 minus cos 𝐴 is equal to tan 𝐴.

So in order to prove this, what we need to do is actually prove that our left-hand side is equal to our right-hand side. So on our left-hand side, like we said we’ve got sin 𝐴 minus two sin cubed 𝐴 over two cos cubed 𝐴 minus cos 𝐴. So the first thing we’re gonna do is we’re actually going to factorize each of these, so factorize the numerator and factorize the denominator.

So first of all, with the numerator, what we’re gonna do is take out a factor of sin 𝐴 because sin 𝐴 is a factor of both sin 𝐴 and two sin cubed 𝐴. And then, inside the bracket, we’re gonna have one minus two sin squared 𝐴. And the reason we’ve got that is because actually sin 𝐴 multiplied by one gives us sin 𝐴. And sin 𝐴 multiplied by negative two sin squared 𝐴 is gonna give us negative two sin cubed 𝐴. Okay, great. So that’s the numerator. Let’s move on to the denominator.

Well, on the denominator, what we can do is actually take out cos 𝐴 as a factor. And then inside the bracket, we’re gonna have two cos squared 𝐴. And that’s because cos 𝐴 multiplied by two cos squared 𝐴 will give us two cos cubed 𝐴. And then we’ve got minus one. And that’s because negative one multiplied by cos 𝐴 gives us negative cos 𝐴.

So now, to actually enable us to actually simplify this further, what we need to do is start using trigonometric identities. And we’re actually gonna use a couple to help us. The first one we’re gonna use is this one here. And this first identity is that sin squared 𝐴 plus cos squared 𝐴 is equal to one. So what we say is sine squared of an angle plus the cosine squared of an angle is equal to one. So how’s this gonna become useful in this question?

Well, it becomes useful because on both the numerator and the denominator we have a one. And therefore, actually using our relationship that we’ve got with our identity, we can see that these ones are actually gonna be equal to sin squared 𝐴 plus cos squared 𝐴. Which is gonna be really useful because actually we’re dealing with sin 𝐴s and sin squared 𝐴s and cos squared 𝐴s as actually part of our question.

So now, what we’re gonna do is actually use our identity. And when we actually substitute in our identity, we’re gonna get sin 𝐴 over cos 𝐴 multiplied by sin squared 𝐴 plus cos squared 𝐴 minus two sin squared 𝐴. And that’s because we actually had a one there which we already said it is equal to sin squared 𝐴 plus cos squared 𝐴. And then all over two cos squared 𝐴 minus, and then again we’ve inserted using our trig identity, we’ve got sin squared 𝐴 plus cos squared 𝐴. And as you noticed, I’ve actually split the fraction into two. So we’ve got the sin 𝐴 over cos 𝐴 separate. And that’s because that would be useful in a minute, using another trig identity.

And now we’re actually gonna simplify. And when we do that, we’re gonna have sin 𝐴 over cos 𝐴 multiplied by cos squared 𝐴 minus sin squared 𝐴 in the numerator. And we got that because we had sin squared 𝐴. And then we’ve got minus two sin squared 𝐴, which gives us negative sin squared 𝐴. And then we’ve got our cos squared 𝐴. So that’s cos squared 𝐴 minus sin squared 𝐴. And then on the denominator, we also have cos squared 𝐴 minus sin squared 𝐴. And the reason we actually get this on the denominator is because we have two cos squared 𝐴.

And then the reason I put in the brackets was obviously to avoid one of the common mistakes here, is we’ve now got minus cos squared 𝐴. And that’s because we had the negative sign in front of the brackets. Which means that the positive in front of the cos squared 𝐴 becomes negative. So we’ve got two cos squared 𝐴 minus cos squared 𝐴 which leaves us with cos squared 𝐴. And then we’ve just got minus sin squared 𝐴.

Okay, great. So now, we’re actually gonna simplify another step further. And we’re gonna do that using another one of our trigonometric identities. And this one is sin 𝐴 over cos 𝐴 is equal to tan 𝐴. So we can actually use that in the answer that we’ve got. So therefore, what we’re gonna get is tan 𝐴. And then just multiply it by one. And the reason it’s multiplied by one is because if you got cos squared 𝐴 minus sin squared 𝐴 over cos squared 𝐴 minus sin squared 𝐴, well this is just equal to one. Because we can actually divide the top and bottom by the same thing. So we can divide the numerator and denominator by cos squared 𝐴 minus sin squared 𝐴.

So therefore, we can say that the left-hand side is equal to tan 𝐴. Which is also equal to the right-hand side because the right hand side was tan 𝐴. So therefore, we can say that we’ve proved that sin 𝐴 minus two sin cubed 𝐴 over two cos cubed 𝐴 minus cos 𝐴 equals tan 𝐴. And we’ve shown all our working here.