First of all, we construct an Isosceles Triangle like this:

With \(AB=BC=a\) (say). And \(B=2x\). Then \(AD \perp BC\) is constructed.

Subsequently, since the angles \(BAC\) and \(ACB\) are equal, angle \(ACB = 90 - x\), then angle \(CAD=x\).

From triangle \(ADC\):

\(\tan \ x\ =\ \frac{CD}{AD}\)

\(\implies \tan \ x\ =\ \frac{a-a\cos 2x}{a\sin 2x}\)

\(\implies \tan \ x\ =\ \frac{1-\cos 2x}{\sin 2x}\)

\(\implies \cos 2x=1-\sin 2x\tan x\)

Squaring both sides,

\(\implies \cos ^22x=1-2\sin 2x\tan x+\sin ^22x\tan ^2x\)

\(\implies 1-\sin ^22x=1-2\sin 2x\tan x+\sin ^22x\tan ^2x\)

\(\implies \sin ^22x=2\sin 2x\tan x-\sin ^22x\tan ^2x\)

\(\sin 2x\) is not **necessarily** \(0\), so we can divide both sides by \(\sin 2x\).

\(\implies \sin 2x=2\tan x-\sin 2x\tan ^2x\)

\(\implies \sin 2x\left(1+\tan ^2x\right)=2\tan x\)

\(\implies \boxed{\sin 2x=\frac{2\tan x}{1+\tan ^2x}}\)

Furthermore,

\(\sin 2x=\frac{2\tan x}{1+\tan ^2x}=\frac{2\tan x}{\sec ^2x}=2\tan x\cos ^2x=2\frac{\sin x}{\cos x}\cos ^2x=\boxed{2\sin x\cos x}\)

Now, from \(\tan \ x\ =\ \frac{1-\cos 2x}{\sin 2x}\):

\(\sin 2x=\ \frac{1-\cos 2x}{\tan x}\)

\(\implies \frac{2\tan x}{1+\tan ^2x}=\ \frac{\left(1-\cos 2x\right)}{\tan x}\)

\(\implies \frac{2\tan ^2x}{1+\tan ^2x}=\ 1-\cos 2x\)

\(\implies \cos 2x=1-\frac{2\tan ^2x}{1+\tan ^2x}\)

\(\implies \boxed{\cos 2x=\frac{1-\tan ^2x}{1+\tan ^2x}}\)

From here,

\(\cos 2x=\frac{1-\tan ^2x}{1+\tan ^2x} = \frac{\cos ^2x\left(1-\tan ^2x\right)}{\cos ^2x\left(1+\tan ^2x\right)} = \frac{\cos ^2x-\sin ^2x}{\cos ^2x+\sin ^2x} = \frac{\cos ^2x-\sin ^2x}{1} = \boxed{\cos ^2x-\sin ^2x}\)

Combining the expansions,

\(\tan 2x=\frac{\sin 2x}{\cos 2x}=\frac{\left(\frac{2\tan x}{1+\tan ^2x}\right)}{\left(\frac{1-\tan ^2x}{1+\tan ^2x}\right)} = \boxed{\frac{2\tan x}{1-\tan ^2x}}\)

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## Comments

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TopNewestAll the more reason to abandon CT in favour of RT... creating a demarcation between \(\sin\), \(\cos\) and \(\tan\) actually makes things 100 times harder, especially when it comes to the situation of "double-angles".

Some criticisms:

If you are going to use visuals, could you at least do them to scale? I cannot see any of the markings on the diagram; seeing as you are clearly using GeoGebra, can you at least use LaTeX to do your labelling? You can also increase the size of the font too.

I think you could be clearer in defining "dropping a perpendicular from A"; these terms should be clear to anyone with a basic understanding of geometry.

There are a few places where you will need further clarification as to your steps; in these places, it would be very hard for someone not familiar with the work to follow through what you are doing.

"\(\sin2x\) is not

necessarily\(0\)" - What if it is zero? I think you need to rephrase it to "assuming \(\sin2x \neq 0\)", and then explaining what happens if \(\sin2x = 0\).Log in to reply

If one looks at the diagram properly, he/she should be able to understand that \(D\) is a point on line segment \(BC\) such that \(AD \perp BC\). I don't feel if this is incomprehensible in any context.

Well, guess what? I've shown this exact post to students who have just learnt elementary trigonometry. Almost half of them don't have an affinity towards maths. Still, every one of them was able to understand it and did not complain about its comprehensibility. I really don't know why you both find this to be

hard to follow.Some things are pretty obvious.One must be able to understand and analyze the cases themselves. Moreover, this post focuses on deriving the double angle formulas using elementary trigonometry, so it mustn't be

elaborating muchon analyzing tiny andobviouscases.I'm sorry if I happen to sound a little rude, but I couldn't help it as others didn't have problems with this.

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Hi Arkajyoti,

Please only think about Gennady as helping you move closer to a more powerful view point.

The key is realizing the power of finite prime fields to explain trigonometry.

I promise that if you explore rational trigonometry... you will have much more powerful tools to help guide students towards not understanding but affinity towards mathematics.

No one understands mathematics. We only pretend to play with the tools of gods.

Lets all pretend and play nicely with these tools, while respecting all voices, even if they are disagreeable.

Actually especially if you find them disagreeable and "rude"... as they may be some of the most powerful and useful ones(at times... not always).

Thanks for you post(as well as Gennady!)

-Peter

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assuming\(\sin 2x \neq 0\) instead of what you wrote still stands. I will also add that in more complicated situations, elaboration and tiny details become significantly more important.Log in to reply

All is to confusing

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I agree. Very poorly written and very hard to follow. I could imagine the average mathematician would struggle to read through this if it was a paper.

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Yeah, it is but an indirect way of deriving the identities using just basic trigonometry.

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Basic "classical trigonometry" is much too transcendental and full of infinite functions. If we use quadrance and spread and pick a specified field we can say much more interesting things!

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As well... this may not be the mathematical "establishment"''s fault. This may be human nature. You often hear, read, or speak about "keeping things straight forward" I know as a teacher my explanations are supposed to be this way.

But this tautological aspect of mathematics that ratios have no linear aspect to them... Is very much the lynch pin. And students are in for a multiplicity of awakenings in this regard(as well as teachers).

It is actually kind of funny if you think about it. No matter how hard you try to suppress this, you will find that it overpowers all aspects of life as we know it.

Students, teachers, administration, etc. cry and complain and argue if you like... this is simply the truth. And it is where the power lies.

The key to getting people onboard is using finite fields to have the rich number theoretic results begin to find their way into popular culture. The issue is someone needs to begin to have them exposed to the light of day.

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i can not understand anything. explanation anyone??????

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Rational trigonometry will help explain everything.

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Actually, I just can’t be stuffed looking or thinking about the question

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The issue is: we all have a lot to learn about this. So get your popcorn, as the show is just about to get interesting.

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Take as much as you can get Annie.

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What did you not understand, the construction of and in the triangle?

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