1. However rich you may be, you can't buy it.

2. However rich you are, you can't buy it.

3. However cold it may be, you must not stay indoors all day.

4. However cold it is, you must not stay indoors all day.

--------------------
Which expressions are correct? Are they all grammatical? Then, which expressions are commonly used?

1. How rich you may be, you can't buy it.

4. How cold it is, you must not stay indoors all day

Jahanara is incorrect.

All 4 of your sentences are fine.

All four expressions are correct and grammatical. However, there are slight differences in meaning and usage between the pairs.

1. "However rich you may be, you can't buy it." - This expression implies that no matter how rich someone might be, they still cannot purchase the mentioned item. The use of "may" suggests a hypothetical or uncertain condition.

2. "However rich you are, you can't buy it." - This expression has a similar meaning to the previous one, but instead of using "may," it uses "are" to indicate a definite or known condition. It suggests that regardless of one's wealth, they still cannot buy the item.

Both expressions 1 and 2 are grammatically correct, but expression 2 ("However rich you are, you can't buy it.") is more commonly used in day-to-day conversations as it represents a more definite condition.

3. "However cold it may be, you must not stay indoors all day." - This expression suggests that regardless of how cold it might be, it is still important not to spend the entire day indoors. The use of "may" implies a possible or uncertain condition.

4. "However cold it is, you must not stay indoors all day." - This expression conveys a similar meaning as the previous one but uses "is" instead of "may" to indicate a definite or known condition. It emphasizes that it doesn't matter how cold it actually is, staying indoors all day is not recommended.

Both expressions 3 and 4 are grammatically correct, but expression 4 ("However cold it is, you must not stay indoors all day.") is more commonly used in everyday language as it represents a more straightforward and assertive statement.