Graph it first.
0 and 2 pi are obvious solutions.
Then substitute 1-cos2t for sin2t and you will convert the equation to a quadratic in is cos x.
Solve the quadratic for the 2 required roots...BUT beware extraneous roots!
-4(1 - cos2t - 3cos t = -3
4 cos2t - 3cos t -1= 0
Now you have a quadratic equation in cos t which gives
(4 cos t +1)(cos t - 1) = 0 so cos t = 1 and cos t = -1/4
That should get it for you.