Subject: simplification of trigonometric expression
From: Richard Fateman
Date: Thu, 01 Nov 2012 18:12:43 -0700
On 11/1/12 6:50 AM, Jean-Claude Arbaut wrote:
> Hello Maxima users,
>
> I am computing the area of the ellipse using the polar equation
> r=p/(1+e*cos(theta)),
> with 0<e<1 (not the simplest way, but I am playing with polar
> coordinates)
> For this, I compute integrate(1/(1+e*cos(theta))^2,theta).
> By hand, I find
> f(e,theta):=2/(1-e^2)^(3/2)*atan(sqrt((1-e)/(1+e))*tan(theta/2))-e/(1-e^2)*(sin(theta)/(1+e*cos(theta)));
>
>
> Maxima finds:
> assume(e>0,e<1)$
> integrate(1/(1+e*cos(theta))^2,theta);
> define('g(e,theta),%);
>
> g(e,theta):=2*(atan(((2*e-2)*sin(theta))/(2*sqrt(1-e^2)*(cos(theta)+1)))/(sqrt(1-e^2)*(e^2-1))-(e*sin(theta))/((cos(theta)+1)*(((e^3-e^2-e+1)*sin(theta)^2)/(cos(theta)+1)^2-e^3-e^2+e+1)))
>
>
> Now, although it's quite obvious by visual inspection or numerical
> samples, I would like to make
> Maxima find that f=g. I was not able to do that with ratsimp, trisimp,
> ...
>
> Is there some function I don't know that could do that ?
>
> Jean-Claude Arbaut
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try onverting to complex exponentials and radcan.