In King Lear, why does Regan need Gloucester's advice to answer letters from Lear and Goneril, since she and her sister have agreed on not to let Lear live with them?
The event that your question refers to occurs at the end of Act II scene 1, when Regan and her husband suddenly appear at Gloucester's residence, and Regan asks for Gloucester's counsel and advice in how to respond to certain letters she has received from her father and sister. Let us just remind ourselves of what she says to Gloucester:
Our father he hath writ, so hath our sister,
Of differences, which I best thought it fit
To answer from our home; the several messengers
From hence attend dispatch. Our good old friend,
Lay comforts to your bosom, and bestow
Your needful counsel to our businesses,
Which craves the instant use.
Perhaps we can infer that Regan is trying to involve Gloucester in this affair to gain his support for what she is about to do in turning against her husband, or perhaps her reference to the "differences" between her sister and father shows that she wants to try and manipulate Gloucester, making him think that she wants to try and make peace between them, whilst all the time she is planning on leaving her father, literally, out in the cold. Certainly we must be aware that both Regan and Goneril are masters of deception and of showing one face whilst secretly hiding another, which should make us suspect their motives.