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Indeed, the default algorithm used by Sage is not capable to find the solutions, but SymPy succeeds:
sage: t = var('t')
sage: solve(cos(2*t) == sin(t), t, algorithm='sympy')
[ImageSet(Lambda(_n, 2*_n*pi + 3*pi/2), Integers),
ImageSet(Lambda(_n, 2*_n*pi + 5*pi/6), Integers),
ImageSet(Lambda(_n, 2*_n*pi + pi/6), Integers)]
