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...

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:

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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.