笃信 is literary and formal - most common with religion, ideology, or principles ('devoutly believes in Christianity / Confucian values / democracy'). For everyday 'I believe you,' use 相信 (xiāngxìn).
memory
笃 (sincere/earnest) + 信 (to believe) = 'earnestly believe.' The 笃 character is also in 笃定 (dǔdìng, certain) and 笃志 (dǔzhì, single-minded).
