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Note that sometimes the "degree reduction" (of the involved trigonometric polynomial) is the wanted and/or needed "simplification".

The corresponding method is called reduce_trig.

The following sample code shows some differences.

sage: var('u');
sage: a = 1 - cos(u)^2
sage: a.simplify_trig()
sin(u)^2
sage: a.reduce_trig()
-1/2*cos(2*u) + 1/2
sage: a.simplify_trig().reduce_trig()
-1/2*cos(2*u) + 1/2
sage: a.reduce_trig().simplify_trig()
sin(u)^2

With an other example...

sage: b = sin(u)^4 - cos(u)^4
sage: b.simplify_trig()
-2*cos(u)^2 + 1
sage: b.reduce_trig()
-cos(2*u)