eqn = {y1'[
t] == (-1 + (3/2) Cos[t]*Cos[t]) y1[
  t] + (-1 - (3/2) Sin[t] Cos[t]) y2[t]}
Solve[eqn, {y2[t]}]

D[(-2 y1[t] + 3 Cos[t]^2 y1[t] - 2 Derivative[1][y1][t])/(
 2 + 3 Cos[t] Sin[t]), t]

eqn2 = {-(((3 Cos[t]^2 - 3 Sin[t]^2) (-2 y1[t] + 3 Cos[t]^2 y1[t] -
    2 Derivative[1][y1][t]))/(2 + 3 Cos[t] Sin[t])^2) + (-6 Cos[
   t] Sin[t] y1[t] - 2 Derivative[1][y1][t] +
 3 Cos[t]^2 Derivative[1][y1][t] - 2 (y1^[Prime][Prime])[t])/(
2 + 3 Cos[t] Sin[t]) == (-1 - (3/2) Sin[t] Cos[t]) y1[
  t] + (-1 + (3/2) Sin[t] Sin[t]) y2[t]}
Solve[eqn2, {(y1^[Prime][Prime])[t]}]