슬퍼하지마 is wrong. 슬퍼하지 마 is correct. 지마 is not a word or a particle. 슬퍼하지 마 is composed with 슬퍼하하(다) + 지 + 마 슬퍼하다 means "to be sad, to grieve, to feel sorry" 지 is a suffix used for denoting negation or prohibition and usually followed by " 않다, 못하다, 말다" 마 is an imperative conjugation form of 말다.(말+아 => 마) 지 마 is used after a verb. 슬프다 is an adjective and 슬퍼하다 is a verb Thus, 슬프지 마 is wrong. e.g. 하지 마 means "don't do it!" 오지 마 means "don't come!" 가지 마 means "don't go!" 먹지 마 means "don't eat!" -아/어하다 is usually used when the speaker observed the emotion or opinion of someone else. Both are correct. 저는 한국에 가고 싶어요. I want to go to Korea 언니는 한국에 가고 싶어해요. I observe that my sister wants to go to Korea. 언니는 한국에 가고 싶어요. = My sister wants to go to Korea. 그는 너무 부끄러웠어요 = He was very shy. It would be weird to say the sentence above, but you might be able to find a sentence like that in a novel or a drama. In novels, authors need to describe the characters in a book, who most of the time are just figments of their own imagination. In the case of adjectives that are not descriptions of one’s feelings, but instead descriptions based on the opinions/facts of somebody/something else, adding ~아/어하다 changes the adjective into a transitive verb. 예뻐하다 is a transitive verb. It requires an object. 그녀는 예뻐해요. is a wrong sentence. 그녀는 예뻐요. She is pretty. 그녀는 강아지를 예뻐해요. She loves a puppy. 그녀는 저를 예뻐해요. She loves me. 나는 슬프다.(O) I'm sad. 나는 슬퍼한다.(?) 그는 슬퍼한다.(O) He is sad. 나는 이별을 슬퍼한다.(O) I'm sad about farewell. (with object) Refer to https://www.italki.com/question/400360

In Korean, virtually all emotion related "adjectives" are reverse-constructed per se. If you knew Spanish, it would be much easier to understand as shown in the example below. https://spanishobsessed.com/blog/like-in-spanish/ 슬프다 actually means "something saddens me". e.g. 아빠가 없는 생일이 슬프다: Birthday without my father's presence saddens me. Please note that birthday is the subject here. 슬프지마 is therefore completely wrong to begin with. 슬퍼하다 on the other hand isn't the re-reversal either. It means "to show signs of sorrow", just like many (almost all) other emotion related adjectives. However, in your case, "슬퍼하지 마" comes quite close to "don't be sad", even if it isn't 100% accurate.

i am korean. and it is my first answer. unfortunately, i can explain logically what you are asking but i think there are some adjective used in two ways like 슬프다 -> 슬퍼하다 귀엽다->귀여워하다 외롭다->외로워하다 이쁘다->이뻐하다 기쁘다->기뻐하다 행복하다->행복해하다