Equation Proof: a*sin2(α) + b*sin(α)cos(α) + c*cos2(α) = (a*m^2 + b*m + c) / m^2 + 1

[rumbata2]

Equation Proof: a*sin2(α) + b*sin(α)cos(α) + c*cos2(α) = (a*m^2 + b*m + c) / m^2 + 1

Ok so basically


if tan(α) = m (α =/= 90 degrees)


Prove that:


a*sin2(α) + b*sin(α)cos(α) + c*cos2(α) = (a*m^2 + b*m + c) / m^2 + 1


a, b, c are parameters




I have tried to replace m with sin(α)/cos(a) and then improve the equation with things like dividing both sides by sinα/cosα, by sinα*cosα but it was completely to no avail and made it even worse to look at. So I pretty much don't know where to begin here but even if someone just points me to the right direction so I can do the rest myself, it would be of great help and greatly appreciated. Anything helps really. Thanks in advance!!

 

[Deleted member 4993]

Ok so basically
if tan(α) = m (α =/= 90 degrees)
Prove that:
a*sin2(α) + b*sin(α)cos(α) + c*cos2(α) = (a*m^2 + b*m + c) / m^2 + 1
a, b, c are parameters

I have tried to replace m with sin(α)/cos(a) and then improve the equation with things like dividing both sides by sinα/cosα, by sinα*cosα but it was completely to no avail and made it even worse to look at. So I pretty much don't know where to begin here but even if someone just points me to the right direction so I can do the rest myself, it would be of great help and greatly appreciated. Anything helps really. Thanks in advance!!

If tan(α) = m → cos(α) = ± 1/√(1+m²) and sin(α) = ± m/√(1+m²)

Use that in the left-hand-side. What do you get?

 

[Harry_the_cat]

Another method is:

Divide LHS by cos^2 (x) * sec^2 (x) ... which equals one when x is not 90 degrees

Replace sec^2 (x) = tan^2(x) + 1 = m^2 + 1

Falls out nicely.

 

[rumbata2]

OP here, little update:

Okay so I read your replies and had trouble at first seeing as how I had to google what sec(x) actually is and had no idea why the stuff you use so freely ( sec^2 (x) = tan^2(x) + 1 and cos(α) = ± 1/√(1+m²) which are essentially the same thing ) is actually true. So anyway I googled sec and managed to write out proof for both of those things by myself (lol not a big accomplishment is it), then used them and did the equation in half a page. Also managed the equation without using any such shenanigans, just the basic trig properties: tg(x) = sin(x)/cos(x) and sin2x + cos2x = 1. Played around with them and expressed the whole LHS with just m. After all, those 2 properties are what you use to prove that sec^2 (x) = tan^2(x) + 1 , so I guess that they are the secret to this whole problem.

Anyway I dunno if what I wrote just now makes any sense but that's not important and I probably sound like an idiot but I'm still in highschool (one that specialises in languages *yawn*) so this stuff is a real eye-opener to me. What's important is, I understood the problem and just solved it in multiple ways, and you guys helped me big time with that. Thank you !!!