. A new treatise on the elements of the differential and integral calculus . - s/ — 1 sin. X, and therefore of all imaginary expres-sions, is thus reduced to an addition, and the raising topowers to a multiplication. 7^. Dividing formula (4) of the preceding article by (o)of the same, member by member, we havee^^~^ _ 2x-^~i — ^^^ ^ + V— 1 sin, a; _ 1 + V— 1 tan. COS. cc — V—Isin. X 1 — \/^Ttan. .t whence, by taking the Napierian logarithms of both members, 2xV^^ = l{l-{- V^l tan. a) - I {i - V^^l tan. .r).Expanding the terms in the second member by Ex. 5, Art. 71, ^ /—- /--r^ , tan.
. A new treatise on the elements of the differential and integral calculus . - s/ — 1 sin. X, and therefore of all imaginary expres-sions, is thus reduced to an addition, and the raising topowers to a multiplication. 7^. Dividing formula (4) of the preceding article by (o)of the same, member by member, we havee^^~^ _ 2x-^~i — ^^^ ^ + V— 1 sin, a; _ 1 + V— 1 tan. COS. cc — V—Isin. X 1 — \/^Ttan. .t whence, by taking the Napierian logarithms of both members, 2xV^^ = l{l-{- V^l tan. a) - I {i - V^^l tan. .r).Expanding the terms in the second member by Ex. 5, Art. 71, ^ /—- /--r^ , tan.^a; . tan.^o; *.r 2x V — 1 — V— 1 tan. x + —^^ V— I —^ j—. 1Q2 DIFFERENTIAL CALCULUS. Equating the imaginary parts in the two members of thisequation, and then dividing through by 2 V— 1, we have tan.^a? , tan.^cc , „ „X 1= tan. X — -\ = ^ \- & — <fec., a series that may be used for the calculation of rt, and whichagrees with the formula in Ex. 6, Art. 71. 75. To find the expansion of cos.^cc in terms of the cosinesof multiples of x. Make e^-^-^ — y; then e«^^^=: ^z, e-^-^^i^-, From formulas 4, 5, Art. 73, we find 1 2 COS. X = e-- + e-»^-i = y + - 2 V — Isin.^^e^-^ _ e--r--i — ^ _-; also, from De Moivres Theorem, we deduce 1 , 1 2 COS. mx = y + —, 2 V— 1 sin. mx = y^^ — —. 1 / 1\ Because + -, 2^ cos.^.r =: f ?/+ - ) : ^ \ y/ but ^ 1\ 77 — 1 , n-1 1 1 1 • +172 r^^ +r^ + r by combining terms at equal distances from the extremes:hence TRIGONOMETRICAL EXPRESSIONS. 103 ; = —--^( COS. nic-|-?i COS. (n — 2)x n — \ \ -\-n COS. (?^ — 4) a:;-|-• • • j {h). / Since there are n
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