If the husband is not the only breadwinner of the family , and the wife is also contributing for the family's income, and she has to take care of the family at the same time shouldn't the husband also take half the role of the housework?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Allah, out of His absolute Wisdom, has granted men guardianship over women and made it a man's duty to maintain his wives and children. Allah Says (what means): {Men are the protectors and maintainers of women, because Allah has made one of them to excel the other, and because they spend (to support them) from their means...} [Quran 4:34] Moreover, the Prophet said: "It is your duty (men) to maintain them (wives) and to clothe them in a reasonable manner" [Muslim]

However, Allah has determined that the best and natural place for the woman is her home. When addressing the Prophet’s wives (including all believing women), Allah Says (what means): {And stay in your houses.} [Quran 33:33]

So, Allah determined the role of men and the role of women in this life. He also obliged men to treat their wives in a good and kind manner; Allah Says (what means): {And live with them in kindness. For if you dislike them — perhaps you dislike a thing and Allah makes therein much good.} [Quran 4:19]

Moreover, the Prophet gave us the best example concerning the treatment of wives. He helped his wives and even did some of the chores in the home as narrated by 'Aa’ishah but he would leave them to the mosque as soon as the Aathaan was called. He used to clean the house and mend his shoes and his dress.

Furthermore, when 'Aa’ishah was asked about the Prophet’s work at home, she said: “He was an ordinary man: he cleans his dress, milks his sheep and serves himself.” [Ahmad, Abu Nu'aym and At-Tirmithi] Another narration reads: “…mends his dress and shoes”. [Ahmad and Ibn Hibbaan] Ibn Hibbaan added: “And mends his bucket.”

Nonetheless, if a husband wants his wife to work outside his home in order to help him in the household, then he should help her in the chores at home; otherwise, this is injustice. In addition to this, if a woman's work outside harms her, she has the right to give it up and it is her husband's duty to maintain her and all the children.

Finally, we advise you to be patient with your husband and do your best in order to please him as the Prophet said: “If a woman performs her five daily prayers, fasts the month of Ramadan, keeps her chastity and obeys her husband, she will enter Paradise.” [Ahmad]

Allah knows best.