kill(all)$
load(atrig1)$
load(ntrig)$

solver=Phil_Todd/Saltire.com$

van=facebook.com/LearningCalculus$

degrees(u):= float((180/%pi)*u);

van=facebook.com/LearningCalculus$

     The Constraint-Generated Equation

geometryExpression:(atan(((sin((%alpha+%phi))*(((sin((%alpha+%phi))*1/2)+
(sin((%alpha+(%phi*(-1))))*(-1/2))+cos((%alpha+%phi))+
(cos((%alpha+(%phi*(-1))))*(-1))))^((-1)))+
(sin((%alpha+(%phi*(-1))))*(((sin((%alpha+%phi))*1/2)+
(sin((%alpha+(%phi*(-1))))*(-1/2))+cos((%alpha+%phi))+
(cos((%alpha+(%phi*(-1))))*(-1))))^((-1)))+
(cos((%alpha+%phi))*(((sin((%alpha+%phi))*1/2)+
(sin((%alpha+(%phi*(-1))))*(-1/2))+cos((%alpha+%phi))+
(cos((%alpha+(%phi*(-1))))*(-1))))^((-1))*1/2)+
(cos((%alpha+(%phi*(-1))))*(((sin((%alpha+%phi))*1/2)+
(sin((%alpha+(%phi*(-1))))*(-1/2))+cos((%alpha+%phi))+
(cos((%alpha+(%phi*(-1))))*(-1))))^((-1))*(-1/2))))*(-1))$

van=facebook.com/LearningCalculus$

     Proof via Phil Todd:

angle:fullratsimp(geometryExpression);

     We want "angle" to be pi/2 radians or 90 degrees
     so the atan is singular, so we set denom to zero
     and solve for the alpha.

fullratsimp(trigexpand(denom(first(first(%)))))=0;

soln:fullratsimp(first(%/2));

     Either phi is 0 (degenerate) or
     "angle" is pi/2. Solving for alpha we have:

first(soln)/sin(%phi) = -second(soln)/sin(%phi);

tan(%alpha) = 1/2;

     Punch atan(1/2) into any calculator to obtain:

degrees(atan(1/2));