kill(all)$

load(ntrig)$
load(atrig1)$

van=facebook.com/LearningCalculus$

a(x):=asin(x)$
b(x):=acos(x)$
f  (x):= a(x) - 1/(b(x) - (1/(a(x) - 1/b(x))))$ f(x)$
fp (x):= ''(diff(f(x),x))$ fp(x)$
fpa(x):= subst(arccos,acos,subst(arcsin,asin,fp(x)))$

limit(f(x),x,0);
newton(t):=float(subst(t,x,x-fp(x)/diff(fp(x),x)))$
maxX:newton(newton(newton(0.8107718004128703)));
f(maxX);
fp(maxX);

s:solve(f(x)=0,x);

g(x):=asin(x)*acos(x)$
h(x):=%pi/2 * asin(x) - asin(x)^2$
[[g(0),h(0)],[g(1/2),h(1/2)],[g(1),h(1)]]$
subst(arccos,acos,subst(arcsin,asin,h(x)))$

trigsimp(solve(h(x)=(3-sqrt(5))/2,x));

float(%);