1!>1 BOOK OF MATHEMATICAL PROBLEMS. 



920. Prove that 



cos 0, cos 20, cos 30, cosn0 



cos20, cos30, cos w0, cos0 



cos30, costf, cos 20 



{cos 0- cos (n 4- 1) 0}"- {1 -cos w0}" 

 2(-l) (l-cosn0) 



cosn0, cos 0, cos 20, ... cos (w - 1) 



921. The product of the roots of the equation 



*-2, 1, 0, 0, 0, 

 1, x-2, 1, 0, 0, 

 0, 1, *-2, 1, 0, 



0, 0, 

 0, 0, 



1, *-2, 1 

 0, 1, a-2 



= 



is n + 1 ; and the sum of the roots is 2n, n being the order of the 

 determinant. 



922. Prove that 



1 1 V 



V, 1, 1 X'j .: 



o, o, o, o, i 



is equal to 1, the second row being formed by differentiating the 

 first with respect to x l9 the third by differentiating the second 

 with respect to # a , and so on. 



