grief, lamentation
歎 centers on expressing deep sorrow or regret: grieving and lamenting aloud. The two given meanings are closely linked as outward expressions of inner pain.
歎 combines 欠 (open mouth, often associated with breath or expression) with a left component that is historically related to 難 or a similar phonetic element, suggesting a vocal expression of grief. The exact historical development of the left side is uncertain.
The right side 欠 looks like a person with an open mouth, and the left side resembles a complex, heavy burden. Picture someone sighing deeply under a heavy load, lamenting their grief.
For タン, imagine a person tanning hides while lamenting their hard work: tan -> タン, and the open-mouthed sigh of grief fits the scene.