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/*
 * agm.maxima
 *
 * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule 
 */

S:  2*a*sin(theta1) / (a+b+(a-b)*sin(theta1)^2);

C2: ratsimp(diff(S, theta1)^2 / (1 - S^2));
C2: ratsimp(subst(sqrt(1-sin(theta1)^2), cos(theta1), C2));
C2: ratsimp(subst(S,                     sin(theta),  C2));
C2: ratsimp(subst(sqrt(1-S^2),           cos(theta),  C2));

D2: (a^2 * cos(theta)^2 + b^2 * sin(theta)^2)
	/
    (a1^2 * cos(theta1)^2 + b1^2 * sin(theta1)^2);
D2: subst((a+b)/2,   a1, D2);
D2: subst(sqrt(a*b), b1, D2);
D2: ratsimp(subst(1-S^2,  cos(theta)^2, D2));
D2: ratsimp(subst(S,      sin(theta),   D2));
D2: ratsimp(subst(1-sin(theta1)^2, cos(theta1)^2, D2));

Q: D2/C2;
Q: ratsimp(subst(x, sin(theta1), Q));

Q: ratsimp(expand(ratsimp(Q)));