To my knowledge, NDSolve doesn't do integro-differential equations.
Solution with NIntegrate:
At first we convert Volterra equation of the first kind to the second kind:
y[x] == D[Sin[x], x]*1/(Cos[x - t] /. x -> t) +
Integrate[D[Cos[x - t], t]/(Cos[x - t] /. x -> t)*y[t], {t, 0, x}](*If (Cos[t,t] is not a Zero*)
(*y[x] == Cos[x] + Integrate[-(Sin[t - x]*y[t]), {t, 0, x}]*)
Then:
a = -3; b = 3; Z = 10;(*On range: -3 to 3*)
Clear[ifunc]
kernel[x_] := Sin[x]
func[x_, 0] := kernel[x]
ifunc[0][x_] := kernel[x]
func[x_?NumericQ, n_Integer] :=
Cos[x] - NIntegrate[kernel[y - x]*ifunc[n - 1][y], {y, 0, x},
MinRecursion -> 4, AccuracyGoal -> 10]
ifunc[j_Integer /; j >= 1] :=
ifunc[j] =
Module[{vals}, vals = Table[{x, func[x, j]}, {x, a, b, 1/Z}];
Interpolation[vals]]
Plot[{1, Evaluate[Table[ifunc[j][x], {j, 0, Z}]][[-1]]}, {x, a, b},
PlotStyle -> {Black, {Dashed, Red}},
PlotRange -> {Automatic, {0, 2}}]
Plot[{1 - Evaluate[Table[ifunc[j][x], {j, 0, Z}]][[-1]]}, {x, a, b},
PlotStyle -> {Black, {Dashed, Red}}](*residuals error*)