# Can you prove this?

#### steveb

Joined Jul 3, 2008
2,436
I was wondering if anyone is able to either prove or disprove the following identity. Please see attached PDF file.

This identity came up in my work recently. I won't go into the details of the application, but I did two different approximate derivations of a function I needed in an engineering application. The two separate functions appeared to give the same numerical answer, but look quite different. One funtion was very simple (y=0.5*x) and the other is much more complicated with an arctan function (see attachment).

My first assumption was that these are probably exactly equal functions; however, when I numerically evaluate the error using double precision calculations in Matlab, I find the error near x=pi/3 to be considerable more than what I expect with double precision. The error is much less as x moves away from this value. Now this error could be due to round off error accumulation, but I'm not convinced of that. (Actually, it probably is due to a zero/zero condition)

I wanted to either prove or disprove this identity, but I've been too busy and can't justify doing this as part of my work since it's just a point of curiousity.

I thought someone here might enjoy tackling this one. I suspect the identity is true, and there is probably a simple elegant way to prove it. I thought of doing out a Taylor expansion, but that looks tedious.

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#### mik3

Joined Feb 4, 2008
4,843
See the attachment, its just application of trigonometric identities.

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#### steveb

Joined Jul 3, 2008
2,436
... its just application of trigonometric identities.
Not just trig identities... Also, a sharp mind!

Great work! I appreciate it very much.

#### Ratch

Joined Mar 20, 2007
1,070
steveb,

Another way is to convert the numerator into all sine terms and the demonminator into all cosine terms. Then apply the difference of sines identity to the numerator and the sum of cosines to the denominator. Everything will cancel except tan(x/2), which is what you are looking for.

Ratch

#### steveb

Joined Jul 3, 2008
2,436
steveb,

Another way is to convert the numerator into all sine terms and the demonminator into all cosine terms. Then apply the difference of sines identity to the numerator and the sum of cosines to the denominator. Everything will cancel except tan(x/2), which is what you are looking for.

Ratch
Yes, that's good too. Thanks.

For me the stumbling block was thinking to represent the constants $${{\sqrt3} \over {2}}$$ and 1/2 as cos and sin of a constant. So simple in hindsight, but I just missed it.

I appreciate the help though. Saves me some time, which is very short lately.

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#### boff1

Joined Oct 15, 2008
26
Clever solution!