Your calculations are incorrect.
There are two sides of the cable around the pulley that helps to support the 80 N load. See the diagrams below. So the vertical force from the cable around the pulley is $F+F\sin(37)$, not $F_y\sin(37)$. This gives the sum of vertical forces as $F+F\sin(37)-80=0$ from which F=49.9 N. (Note that $F_y\sin(37)$ does not make sense.)
The horizontal force from the cable around the pulley is $F\cos(37)$, not $F_x\cos(37)$. Furthermore, there is a force in the horizontal "rope" of magnitude T, so the sum of forces in the horizontal forces is $F\cos(37)-T=0$ from which T=39.9N.
Here is a free body diagram showing all of the forces acting on the pulley. The statement to neglect the friction in the pulley indicates that the tension in the cable is the same value, F, on each side of the pulley.
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