Generally you do not have to remove the dead grass when reseeding the dead spots.  Simply scatter grass seed in the dead spot and make sure it falls through the dead grass and makes contact with the soil.  The dead grass will actually shade the new seed reducing the possibility of damage from the sun.  If the dead grass is so dense that the seed does not drop through it to make contact with the soil you can fluff it up so the seed will drop through it.

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