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Not directly in Sage or packages included in Sage :

sage: var("t")
t
sage: Sol=solve(cos(2*t)==sin(t), t, to_poly_solve=True)
sage: Sol
[t == -1/2*pi - 2*pi*z0, t == 1/6*pi + 2/3*pi*z1]

We have to declare the integer constants:

sage: var("z0, z1", domain="integer")
(z0, z1)
sage: assumptions()
[z0 is integer, z1 is integer]

However, neither Maxima nor Sympy can use the domain constraint to find the relevant solutions. For the first solution :

sage: solve([Sol[0].rhs()>=0, Sol[0].rhs()<=pi/2], z0)
[[(-1/2) < z0, z0 < (-1/4)], [z0 == (-1/2)], [z0 == (-1/4)]]
sage: solve([Sol[0].rhs()>=0, Sol[0].rhs()<=pi/2], z0, algorithm="sympy", domain=
....: "integer")
[[(-1/2) <= z0, z0 <= (-1/4)]]

and Sage has no direct way to find that the intersection of these intervals (both [-1/2 -1/4] with $\mathbb{N}$ is the empty set.

For the second solution:

sage: solve([Sol[1].rhs()>=0, Sol[1].rhs()<=pi/2], z1)
[[(-1/4) < z1, z1 < (1/2)], [z1 == (-1/4)], [z1 == (1/2)]]
sage: solve([Sol[1].rhs()>=0, Sol[1].rhs()<=pi/2], z1, algorithm="sympy", domain=
....: "integer")
[[(-1/4) <= z1, z1 <= (1/2)]]

The admissible set of solutions for z1 is, in both cases, [-1/4 1/2], whose intersection with $\mathbb{N}$ (that Sage cannot directly compute) is {0}, giving us the solution :

sage: Sol[1].subs(z1==0)
t == 1/6*pi

Note that this is one case where Mathematica can do the job more efficiently:

sage: mathematica.Solve([cos(2*t)==sin(t), t>=0, t<=pi/2], t)
{{t -> Pi/6}}

In Sage, objects representing infinite sets are available. One might try to play with them to obtain "better" representations of the solutins, but this is not (yet) implemented.

HTH,

Not directly in Sage or packages included in Sage :

sage: var("t")
t
sage: Sol=solve(cos(2*t)==sin(t), t, to_poly_solve=True)
sage: Sol
[t == -1/2*pi - 2*pi*z0, t == 1/6*pi + 2/3*pi*z1]

We have to declare the integer constants:

sage: var("z0, z1", domain="integer")
(z0, z1)
sage: assumptions()
[z0 is integer, z1 is integer]

However, neither Maxima nor Sympy can use the domain constraint to find the relevant solutions. For the first solution :

sage: solve([Sol[0].rhs()>=0, Sol[0].rhs()<=pi/2], z0)
[[(-1/2) < z0, z0 < (-1/4)], [z0 == (-1/2)], [z0 == (-1/4)]]
sage: solve([Sol[0].rhs()>=0, Sol[0].rhs()<=pi/2], z0, algorithm="sympy", domain=
....: "integer")
[[(-1/2) <= z0, z0 <= (-1/4)]]

and Sage has no direct way to find that the intersection of these intervals (both [-1/2 -1/4] with $\mathbb{N}$ is the empty set.

For the second solution:

sage: solve([Sol[1].rhs()>=0, Sol[1].rhs()<=pi/2], z1)
[[(-1/4) < z1, z1 < (1/2)], [z1 == (-1/4)], [z1 == (1/2)]]
sage: solve([Sol[1].rhs()>=0, Sol[1].rhs()<=pi/2], z1, algorithm="sympy", domain=
....: "integer")
[[(-1/4) <= z1, z1 <= (1/2)]]

The admissible set of solutions for z1 is, in both cases, [-1/4 1/2], whose intersection with $\mathbb{N}$ (that Sage cannot directly compute) is {0}, giving us the solution :

sage: Sol[1].subs(z1==0)
t == 1/6*pi

Note that this is one case where Mathematica can do the job more efficiently:

sage: mathematica.Solve([cos(2*t)==sin(t), t>=0, t<=pi/2], t)
{{t -> Pi/6}}

In Sage, objects representing infinite sets are available. One might try to play with them to obtain "better" representations of the solutins, but this is not (yet) implemented.

EDIT : Sympy can do that, but using sympy's sets, whoch are not (yet) backtranslatable to Sage's Sets :

 t=var("t")
Eq=cos(2*t)==sin(t)
import sympy
SS=solve(Eq, t, algorithm="sympy"); SS

[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)]

Note that tis solution is a lis of sympy objects, not directly backtranslatable to sage:

[type(u) for u in SS]
[<class 'sympy.sets.fancysets.ImageSet'>,
 <class 'sympy.sets.fancysets.ImageSet'>,
 <class 'sympy.sets.fancysets.ImageSet'>]

S1=reduce(lambda a,b:a.union(b), SS,
          sympy.EmptySet).intersection(sympy.Interval(0,pi/2))

This solution is again a sympy set, not directly backtranslatable to Sage :

S1
FiniteSet(pi/6)
S1._sage_()
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-173-5ca5ebb7fb66> in <module>()
----> 1 S1._sage_()

AttributeError: 'FiniteSet' object has no attribute '_sage_'

A wokaround is to manually backtranslate :

Sol=Set([u._sage_() for u in S1.args]) ; Sol
{1/6*pi}

HTH,