Here is one way... not sure it is the best
sage: eq1 = sqrt(cos(4*x)+1)
sage: eq2 = eq1.simplify_trig()
sage: eq2
sqrt(8*cos(x)^4 - 8*cos(x)^2 + 2)
The next step consists in factoring what is inside the sqrt:
sage: o = eq2.operands()[0]
sage: of = o.factor()
sage: o
8*cos(x)^4 - 8*cos(x)^2 + 2
sage: of
2*(2*cos(x)^2 - 1)^2
sage: eq3 = eq2.subs({o:of})
sage: eq3
sqrt(2)*sqrt((2*cos(x)^2 - 1)^2)
Then for the conversion sqrt(x^2) -> abs(x) you can use
sage: eq3.simplify_real()
sqrt(2)*abs(2*cos(x)^2 - 1)
Vincent
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