Final answer:
James K. Polk obtained his cabinet's support for war by strategically sending troops to provoke a conflict, using the idea of Manifest Destiny, and capitalizing on Congress's support bolstered by anti-Mexican sentiment and the desire for territorial expansion.
Step-by-step explanation:
President James K. Polk was instrumental in garnering full support from his cabinet for a declaration of war against Mexico, which was prompted by disputes over Texas's southern border and Polk's desire for western expansion. The tactics used by Polk included provocation through his envoy in Mexico, raising tensions that led to the stationing of U.S. troops at the Rio Grande, claimed by Texan expansionists as the border. A skirmish ensued between General Zachary Taylor's forces and Mexican troops, leading to Polk's war declaration on the grounds that Mexico had invaded U.S. soil. Despite some opposition from Whigs like Abraham Lincoln, who demanded clarification on where the incident occurred, the broader Congress supported the war, influenced by beliefs of Mexican weakness and anti-Mexican sentiment prevalent at the time.
Furthermore, the enthusiasm for the war was bolstered by the belief in Manifest Destiny and a strong antislavery opposition that criticized the war as a means to annex more slave territory. Lincoln, as part of the antislavery faction, issued the "Spot Resolutions" to question the legitimacy of Polk's claims, but the Democrats, along with volunteer forces, supported Polk, tipping the scales in favor of war. It was a combination of strategic moves, political support from Democrats, and the widespread idea of America's right to expansion that allowed Polk to win over his cabinet and Congress for the war effort against Mexico.