Silicone caulk - check the tube at the hardware store, as it will tell
you what sufaces to use it on. Corners - wall-to-wall and to ceiling
and floor - will always flex, so should be treated with something that
is flexible and can expand/contract a little more than grout can. Be
sure to clean the corner immaculately - scrub off all soap scum or
deposits and then wipe down (and let dry, of course) with full-strength
bleach so mold will not grow under the new caulk.
Is that a corner mortar joint, if so it indicates yearly house
shifting id guess. I would clean , grout, then caulk it since it will
still shift. Clean well since its full of soap, lime residue and mold,
a hand grout saw will get you new grout.
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