The remainder is R(2m+ 1), which is expanded into 2m terms. Because the expansion contains x^(2m term), the latter term is obviously 2m+ 1 power of X, so it is R(2m+ 1). Derivation of remainder: Is it the first equal sign or the second equal sign? The first equal sign is directly obtained according to Taylor's mean value theorem, and the second equal sign, like the last one, is an identity transformation of trigonometric functions. Use the formula COS (a+kπ) = (- 1) k * COSA.