# Trigonometric simplifications and matrices

I want to simplify trigonometric identities in a matrix. For example, say I want to show that the composition of two rotation matrices is a rotation, I can do with sage something like

  var("theta1,theta2")
Rtheta1=column_matrix([[cos(theta1),sin(theta1)],[-sin(theta1),cos(theta1)]])
Rtheta2=column_matrix([[cos(theta2),sin(theta2)],[-sin(theta2),cos(theta2)]])
produit=Rtheta1*Rtheta2
show(produit.simplify_trig())
show(produit.apply_map(lambda x: x.trig_reduce()))


Note that simplify_trig or trig_reduce don't work on matrices and that you need to use apply_map to use it entry by entry, as detailed in Mike Hansen's answer in this question .

However when I get to 3 matrices, sage can't simplify with the above procedure:

var("theta1,theta2,theta3")
Rtheta1=column_matrix([[cos(theta1),sin(theta1)],[-sin(theta1),cos(theta1)]])
Rtheta2=column_matrix([[cos(theta2),sin(theta2)],[-sin(theta2),cos(theta2)]])
Rtheta3=column_matrix([[cos(theta3),sin(theta3)],[-sin(theta3),cos(theta3)]])
produit=Rtheta1*Rtheta2*Rtheta3
show(produit.apply_map(lambda x: x.trig_reduce()))


For example, the 1-1 entry in this matrix is returned as cos(theta1 + theta2)*cos(theta3) - sin(theta1 + theta2)*sin(theta3) . The weird thing is that using (cos(theta1 + theta2)*cos(theta3) - sin(theta1 + theta2)*sin(theta3)).trig_reduce() produces the correct simplification cos(theta1 + theta2 + theta3) .

What's happening here? Any other way to force the simplification?

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Apparently trig_reduce is not idempotent and you have to apply it twice to get the correct result:

sage: produit.apply_map(lambda x: x.trig_reduce().trig_reduce())
[ cos(theta1 + theta2 + theta3) -sin(theta1 + theta2 + theta3)]
[ sin(theta1 + theta2 + theta3)  cos(theta1 + theta2 + theta3)]

more

Wow, what is even more weird is that it seems that if you go with an even number of matrices, sage can simplify with one trig-reduce, but needs two for an odd number of matrices