Well, ultimately divide and rule is done to serve a greater political or strategic aims. You could make a case that the support of Muslim League by Britain,, was begun in earnest. once it became clear that the Congress Party,under Gandhi and Nehru, would not subordinate the strategic interests of an independent India to the West.
A Pakistan,allied to the West and with a powerful military, served as critical buffer state between the USSR, China and India. Indeed Pakistan historically is one of the most useful allies to the US.
As we know, the US nearly went to full-scale war with India to save Pakistan in 1971 and it was only the Soviet Navy that prevented this. The US ultimately had to join forces with China, to preserve (west) Pakistan's territorial integrity. Pakistan's collapse in 1971 and Americas détente with China meant the US was beholden to China and,so, that a complete Cold War victory for the US in Asia, including a win in Vietnam was no longer possible
China was given permanent membership of security council ,with US support, in September 1971,taking the seat formerly occupied by Taiwan . Pakistan surrendered to India in December 1971 and the eastern wing of Pakistan broke away to form Bangladesh.
"The PRC also replaced Taiwan (Republic of China) as a permanent member of UN Security Council in 1971.
Its first veto was used to support Pakistan during the Indo-Pakistani War of 1971.
In the early years of Bangladesh's independence, Dhaka was also close to the Soviet Union, which was a rival of China following the Sino-Soviet split.
Despite the support of most countries in the world, Bangladesh's UN membership was vetoed by China until 1974. "
https://en.wikipedia.org/wiki/Bangladesh–China_relations
A little thinking will make ,it will be clear why a violation of Pakistan's territorial integrity would've been a strategic red line for both the US and China.
America didn't want Soviet forces reaching the Arabian Sea and so threatening the global supply of gas and oil in the Middle East.
China(and this might be relevant to the thread) didn't want the Soviet and Indian armies linking up and threatening its soft underbelly , the Tibetan Plateau,from which the headwaters of major rivers in Pakistan ,Northern India, South-East Asia and China flow.