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Yes, it is possible.
Doubling a cube Trisecting any angle
No. This is known to be impossible. For more information, including a proof, check the Wikipedia article on "doubling the cube".
Doubling a cube and trisecting any angle
The Delian Problem, doubling the cube, is considered one of the most famous impossible problems from Greek antiquity. Two others are the trisection of the general angle and the squaring of a circle.