g-SO DEMONSTRATION OF THE COTESIAN THEOREM. 



Demonstration Or making n — 1 — m, and transposing, >ve have 



of Cotes's 



theorem. o cos . x X 2 cos. rti x = 2 cos. (rai — 1) x -f 2 cos. (w + 1 ) x. 



When, by making 2 cos. x — y -\ , and admitting, that 



in any case, 2 cos. {m — 1) x — y m - » -\ — - — , and 2 cos. 



mx~y m -\ — -, (which we know to be true when m~2 



from the above forms), we shall find that 2 cos. (m + 1) *'— 



«m + i _j • that is, it will be of the same form with re- 



gard to the multiple m •+- 1, which may be shown as fol- 

 lows. 



In the foregoing general formula, by substituting for 

 2 cos. x, 2 cos. (»j — 1) x, and 2 cos. m x, their respective 



values w -1 , y m - ' + -, and w m -J ; we have, 



y y m ~ y m 



2cos.(m+l)x = (^+^)x( i / + y)-(^- 1 + ^r:) 

 the latter side of which equation being reduced gives 



2 cos. (m+l),i; = i/ m + H 



that is, it is of the same form with regard to its multiple 

 as the two preceding forms; and since we know that those 

 are true when m ~ 2, or when mzzl, it follows, from what 

 is said above, that it is true when m — o, and consequently 

 also when m — 4, and generally for any value of m. We 

 may therefore conclude with certainty, that if 



1 



2 cos. x — y -\ 



V 



that 2 cos. m x — y m \ 



y m 



From which we readily deduce the following equations: 



1st, y* — 2 y cos. x + 1 — 



2d, y°~ m . — 2 y™ cos. m x + 1 — 



Now 



