To some extent, "divide and rule" is a kind of post-facto description of common policies adopted by empires (and other occupiers) that ascribes conscious, deliberate intention on the part of imperial authorities.

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As a description of the consequences of typical imperial policies--in this case, of the modern British Empire in South Asia, Africa and the Middle East--"divide and rule" is completely accurate. In all of its modern imperial possessions, Great Britain defined its subjects through formalized ethnic, racial and religious identifiers and associated them with specific locations and territories, usually constraining them to some degree to live and work in the locations that had been designated for their group. These processes of formal definition typically exaggerated the distinctions between existing groups and identities in colonial societies, invented or fabricated attributes or characteristics of those groups, and built in the presumption that different groups would be antagonistic or rivalrous with one another but for the intervention of British imperial authority. Moreover, the British (as well as other European imperial powers in their possessions) frequently privileged some groups over others in various ways--say, in allowing some groups to serve as low-level bureaucrats or employees within the imperial administration, or relying heavily on particular groups to staff police or military units. At the same time, other groups often were subject to particularly harsh forms of imperial intervention, perhaps because they strongly contested the ways in which their identities were defined or the locations to which they were assigned, or because imperial policy sharply intervened in their economic lives in extremely negative ways.

Nigeria is often cited as a classic African example of British "divide and rule". First, the British defined subjects within the territory formally as "Yoruba", "Igbo", "Hausa". People who had parents or kin in two or more groups (a substantial number of people) was ultimately arbitrarily designed as exclusively one of those groups. People who did not identify strongly with these identity markers or whose cultural and linguistic lives incorporated multiple traditions and social practices (as did many people living in the borders between Yoruba and Igbo communities, for example) were sorted into sharply formal and separated identities as far as the imperial state was concerned. Differences and distinctions within these "ethnic" categories were ignored by imperial administrations. The British imposed standard formal administrative systems on all "tribes" in their African territories, including the appointment of "chiefs"--but Igbo political traditions did not include the concept of chief in this sense. And yet the British forced them to have "chiefs". The imperial administration in Nigeria assumed that 'tribes' were antagonistic towards one another and said as much regularly, while it frequently favored northern Hausa/Islamic communities. Smaller groups in the Cross Rivers region were often ignored or overlooked in favor of the larger ethnic identities. Etc.

Some of these "divide and rule" policies were also imposed across the empire as a whole. The British very consciously responded to political disturbances or uprisings within their empire with troops drawn from far-away colonies, for example. These kinds of strategies are perhaps where the conscious intention to "divide and rule" were most clearly visible in the deliberations of various imperial leaders and policy-makers. But it is worth remembering more generally that in the 19th and early 20th Century, many officials in the Colonial Office, many British military officers, and at least some politicians had considerable exposure to the history of classical Greece and Rome in their educational training, and moreover had read important Western thinkers on civics and politics like Machiavelli. The doctrine of "divide and rule" was an old idea that they knew about from those sources. It is not hard to infer that many of them were consciously aware of the practical advantages of segregating imperial subjects into small groups and emphasizing the separations and antagonisms between those groups. (And equally of the dangers should imperial subjects come to see themselves as part of one large group or identity who shared a common interest in overthrowing imperial control.)

On some occasions, you don't have to infer: you can find officials and leaders more or less saying that this is their intention. More typically, however, British officials acted as if they really believed that Hindu, Muslim, Sikh and Parsi were completely separable and antagonistic identity categories in South Asia, or that Igbo, Hausa, Yoruba, Ibibio etc. "tribes" were completely different groups who had long-standing enmities with each other. Colonial civil servants were trained to think this way. Because these categories were so strongly formalized as part of imperial rule, they gained power and currency over time--in effect, making 'divide and rule' a successful strategy.

The only caution I'd add, however, is that the common assertion by many nationalist intellectuals and politicians in India that Hindu and Muslim divisions were 100% a creation of British rule (and similar assertions elsewhere throughout the postcolonial world) is too strong a claim. Formalizing these identities as a basic principle of imperial rule had a huge impact but it did not create such identities from nothing, nor was everything before imperial rule entirely peaceful and collaborative between different religious, linguistic and cultural groupings of people.

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In the case of British India, "Divide and Rule" is a retroactive simplification of a long (335 year) process of economic, political and military flux. At worst, "Divide and Rule" is an accusation of premeditation for conquest and exploitation on the part of the East India Company and the the British Raj.

When the East India Company first arrived in what we now call India, it became yet another player in an incredibly crowded political, cultural, religious, racial and geographically diverse landscape comparable in size to Western and Central Europe combined. The majority of the Indian subcontinent was held together by a patchwork of Muslim Mughal rulers, princes and their officials. The East India company, for complex economic, political and military considerations was drawn into a piecemeal appropriation of regions and alliances. As conflicts and succession disputes arose in what was a weakening Mughal political structure, the Company capitalised on them. It's motivations centred around trade, tariff free exchange of goods across borders and protection of both. The East India Company took advantage of divisions whenever it's interests were threatened or could be expanded. There was no need to create divisions as they existed in abundance. From 1784 after Pitt's Act which put the Company under the control of the British Government, exacerbating divisions within the Indian populous became antithetical to Company and British Government interests.

In essence, divisions were exploited when it was advantageous to do so. However for the most part it was not advantageous to do so. After the mutiny of 1857 which was largely whipped up by religious rumour and racial tension, the British made ongoing efforts to combat division in favour of unity and harmony.

Perhaps one origin for the "Divide and Rule" claim regarding India is Winston Churchill who was not in favour of Indian Independence so soon. Although a rare genius of a man in many regards, he was not without his faults and failures. His opposition to Indian Independence is one such failure and expressed that exploiting religious tension between Hindus and Muslims was an opportunity to maintain British rule in India.