By I. J. Schwatt

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A y = cotx. I. if 1 ( - 1)a G-a) a2B - (87) jg. fix" y=i+^TTi < 88 > DERIVATIVES OF TRIGONOMETRIC FUNCTIONS But, by Ch. I. (83), -^ k * d«yt dx (^gS(-ir(a)an 1 uk N W (w + i sin a;)* " 1 (cos x - l)^ 1 ~ (90) l; ( 1 2*+ 1 i + 1 sin 2 x sin*" 1 z fc = ~ 9^i cosec2a: ( 1 -icotx)*- 1 and where 2 ^'^^b) 00^ w . x g £g (-l) a (Ja^W 2 ,. (96) and (96) gives rft-2+y l |-cots=(-l)L* J2«eosec^2^S(- 1 a S n C)« ) k-l 2/5+1-y (ii) l-(-D y= . where We shall cot2P+i-y, (97) n ^ next find the expansion of y = xcotx.

S(»} (18) i ; . OPERATIONS WITH SERIES 44 2(»)-5(a. (Sko-tta. )». But (1 + i)» = (V2)»(^2 + i-ff +*)" (1 (25) (26) = (s/2)» (cos j + isin = (v 2)»e*. , \n / Similarly (1 - i) n and Now = ( v/2) ne""r = (^2)" (cos ^ - i sin Then, by means of (27) and — 008^ = ^-^ =(- . (28) obtain from (25) and (26), 5=( v/2)«cos^ (29) ^(^fsm— (30) 1 f~— 1 4 J 1)L we (28), (27) Wirt + ( — 1)2 — 1 , , when n is even, (31) when n is odd (32) _ 1 SERIES OF BINOMIAL COEFFICIENTS cos^ = therefore \ [( 1)R^2 { 1 -(- 1)»} 45 + ( - l)GG jl + ( {l + (-l) n }} (33) whether n be even or odd.

2n We then obtain 2/ = 2(-l)* (*•)' 2. (i) To y = tan Given find and the expansion Now and of y= ~= dxn 2t-rdx n y. 2i ,2ix + =-, u+V 28 -I, I where u = e 2ix . , DERIVATIVES OF TRIGONOMETRIC FUNCTIONS Then, by Ch. I. (83), ~1 uk g^j. g(-irC)rf (11) and 2n+1 1 d 2n+1 (-l)n+i22«+i ^sec^cos(^-l)a TI tan^ = ^ 2 A=l - Combining (11) and /K fc ; £(-l) a r) a 2 ^i. )« /3 Now and tan a; = d2n V , (14) (12), (15) ~~\ ^^tanzj since — tana; . C )*-ig(-l)"g) a-. x i J 2« sec^g ^ pg < - 1)' n («) + . z we find the expansion of tanir, x let =0 in (24), finally (17).

### An Introduction to the Operations with Series by I. J. Schwatt

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