tion. 



DEMONSTRATION OF THE COTESIAN THEOREM. 279 



either within the circle* or in the diameter produced, and 

 join PB, PC, PD, &c.,then will 



PBxPDxPF &c. rAO' + PO" 1 



and P A x P C xPEkrAO»j3 P O m 



Demonstration. 

 It is a well known trigonometrical property* that by Demonstra- 

 makiug cos. x —p> we may derive, 



cos. x — p 



cos. 2 x — 2 f» a — 1 



cos. 3 x =r 4 p 3 — 3 p 



cos. 4x rz 8/-8^+ 1 



cos. 5xz~ l6p s — 20 p 3 + 5 p 



&c. &c. See Bonnycastle's Trig, p. 301, 



Now by substituting 2 p zz y -\ , and multiplying 



y 



each of the above formulae by 2, they are reduced to the 

 following simple forms: 



2 cos. x zz y -\ 



y 



2 cos. 2 x zz y* -f -=- 



y 



. 1 

 2 cos. 3 x — y * + — r 



2 cos. 4i = ^+ — 



2 cos. 5xzz y* -\ r 



whence we may con- 7 m 1 



elude generally j 2 cos. «x=y» + ^ 



But as this general form is only deduced from observing 

 the law of the leading forms, it will be more sa isfactory to 

 see it derived in a direct manner : which may be done by 

 means of the general formula* 



Cos. n x zz 2 cos. x . cos. [n — 1 ) x— cos. (n ~ 2) x. 



Bonnycastle's Trig. p. 300. 



Or 



