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630. 
ON AN EXPRESSION FOR 1 ± sin (2p +1) w IN TERMS OF sinw. 
[From the Messenger of Mathematics, vol. v. (1876), pp. 7, 8.] 
WRITE sin u — x, then we have 
sin u — x, cos u = V(1 — x' 1 ), 
sin %u — 3x— 4ox, cos 3 u— (1 — 4a; 2 ) \/(l — x 2 ), 
sin 5u = 5x — 20^ + 16a; 5 , cos 5m = (1 — 12a; 2 +16a?*) \/(l — ¿¡O, 
&c. &c. 
It is hence clear, that in general 
1 — sin (2p + 1) u = (1 ±x) {(1, xY\ 2 , 
1 + sin (2p + 1) u = (1 + x) {(1, — x)p} 2 , 
where (1, x) p denotes a rational and integral function of x of the order p, and 
(1, — x) v the same function of — x\ for it is only in this manner that we can have 
cos 2 (2p + 1) u = (1 — x 2 ) {[1, # 2 p} 2 . 
We, in fact, find 
1 + sin u = 1 + x, 
1 — sin 3u = (1 + x) (1 — 2a;) 2 , 
1 + sin 5u = (1 4- x) (1 + 2x — 4x 2 ) 2 , 
1 — sin 7u = (1 + x) (1 — 4x — 4a? + 8ar*) 2 , 
&c. 
and it thus appears that the form is 
1 + (—) v sin (2p + 1) u = (1 + x) {(1, #)p} 2 . 
C. X. 
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